pub fn main() {
    crate::prepare!();
    for _ in 0..sv!(usize) {
        sc!(n, m, s: Bytes, p: Bytes);
        let res = wildcard_pattern_matching(&p, &s);
        R::init_with_time(n + m);
        let hs = R::hash_sequence(
            s.iter()
                .map(|&s| if s == b'?' { b'a' as _ } else { s as _ }),
        );
        let hp = R::hash_sequence(p.iter().map(|&s| s as _));
        let mut ans = None;
        for i in 0..=n - m {
            if res[i] {
                if let Some(ans) = &mut ans {
                    let j = *ans;
                    let xs = vec![hs.range(0..i), hp.range(..), hs.range(i + m..n)];
                    let ys = vec![hs.range(0..j), hp.range(..), hs.range(j + m..n)];
                    if HashedRangeChained::from(xs) < HashedRangeChained::from(ys) {
                        *ans = i;
                    }
                } else {
                    ans = Some(i);
                }
            }
        }
        if let Some(i) = ans {
            let ans = s[..i]
                .iter()
                .map(|&s| if s == b'?' { b'a' } else { s })
                .chain(p)
                .chain(s[i + m..].iter().map(|&s| if s == b'?' { b'a' } else { s }))
                .map(|b| b as char);
            pp!(@ns@it ans);
        } else {
            pp!(-1);
        }
    }
}
type R = Mersenne61x1;
pub fn wildcard_pattern_matching(p: &[u8], s: &[u8]) -> Vec<bool> {
    let n = p.len();
    let m = s.len();
    assert!(n >= 1);
    assert!(m >= 1);
    let mut sum = vec![M::zero(); m - n + 1];
    macro_rules! add {
        ($f:expr, $g:expr) => {{
            let f = $f;
            let g = $g;
            let mut x = vec![M::zero(); n];
            let mut y = vec![M::zero(); m];
            for i in 0..n {
                x[i] = f(p[n - i - 1]);
            }
            for j in 0..m {
                y[j] = g(s[j]);
            }
            let z = Convolve998244353::convolve(x, y);
            for i in 0..=m - n {
                sum[i] += z[n + i - 1];
            }
        }};
    }
    add!(|x| M::from((x != b'?') as u32 * x as u32 * x as u32), |x| {
        M::from((x != b'?') as u32)
    });
    add!(
        |x| -M::from((x != b'?') as u32 * x as u32 * 2),
        |x| M::from((x != b'?') as u32 * x as u32)
    );
    add!(|x| M::from((x != b'?') as u32), |x| M::from(
        (x != b'?') as u32 * x as u32 * x as u32
    ));
    sum.into_iter().map(|s| s.is_zero()).collect()
}
pub type M = montgomery::MInt998244353;
#[allow(unused_imports)]use std::{cmp::{Ordering,Reverse},collections::{BTreeMap,BTreeSet,BinaryHeap,HashMap,HashSet,VecDeque}};
mod main_macros{#[doc=" Prepare useful macros."]#[doc=" - `prepare!();`: default (all input scanner (`sc!`, `sv!`) + buf print (`pp!`, `dg!`))"]#[doc=" - `prepare!(?);`: interactive (line scanner (`scln!`) + buf print (`pp!`, `dg!`))"]#[macro_export]macro_rules!prepare{(@output($dol:tt))=>{#[allow(unused_imports)]use std::io::Write as _;let __out=std::io::stdout();#[allow(unused_mut,unused_variables)]let mut __out=std::io::BufWriter::new(__out.lock());#[allow(unused_macros)]#[doc=" [`iter_print!`] for buffered stdout."]macro_rules!pp{($dol($dol t:tt)*)=>{$dol crate::iter_print!(__out,$dol($dol t)*)}}#[cfg(debug_assertions)]#[allow(unused_macros)]#[doc=" [`iter_print!`] for buffered stderr. Do nothing in release mode."]macro_rules!dg{($dol($dol t:tt)*)=>{{#[allow(unused_imports)]use std::io::Write as _;let __err=std::io::stderr();#[allow(unused_mut,unused_variables)]let mut __err=std::io::BufWriter::new(__err.lock());$dol crate::iter_print!(__err,$dol($dol t)*);let _=__err.flush();}}}#[cfg(not(debug_assertions))]#[allow(unused_macros)]#[doc=" [`iter_print!`] for buffered stderr. Do nothing in release mode."]macro_rules!dg{($dol($dol t:tt)*)=>{}}};(@normal($dol:tt))=>{let __in_buf=read_stdin_all_unchecked();#[allow(unused_mut,unused_variables)]let mut __scanner=Scanner::new(&__in_buf);#[allow(unused_macros)]macro_rules!sc{($dol($dol t:tt)*)=>{$dol crate::scan!(__scanner,$dol($dol t)*)}}#[allow(unused_macros)]macro_rules!sv{($dol($dol t:tt)*)=>{$dol crate::scan_value!(__scanner,$dol($dol t)*)}}};(@interactive($dol:tt))=>{#[allow(unused_macros)]#[doc=" Scan a line, and previous line will be truncated in the next call."]macro_rules!scln{($dol($dol t:tt)*)=>{let __in_buf=read_stdin_line();#[allow(unused_mut,unused_variables)]let mut __scanner=Scanner::new(&__in_buf);$dol crate::scan!(__scanner,$dol($dol t)*)}}};()=>{$crate::prepare!(@output($));$crate::prepare!(@normal($))};(?)=>{$crate::prepare!(@output($));$crate::prepare!(@interactive($))};}}
pub use self::iter_print::IterPrint;
mod iter_print{use std::{fmt::Display,io::{Error,Write}};pub trait IterPrint{fn iter_print<W,S>(self,writer:&mut W,sep:S,is_head:bool)->Result<(),Error>where W:Write,S:Display;}macro_rules!iter_print_tuple_impl{(@impl$($A:ident$a:ident)?,$($B:ident$b:ident)*)=>{impl<$($A,)?$($B),*>IterPrint for($($A,)?$($B),*)where$($A:Display,)?$($B:Display),*{#[allow(unused_variables)]fn iter_print<W,S>(self,writer:&mut W,sep:S,is_head:bool)->Result<(),Error>where W:Write,S:Display{let($($a,)?$($b,)*)=self;$(if is_head{::std::write!(writer,"{}",$a)?;}else{::std::write!(writer,"{}{}",sep,$a)?;})?$(::std::write!(writer,"{}{}",sep,$b)?;)*Ok(())}}};(@inc,,$C:ident$c:ident$($D:ident$d:ident)*)=>{iter_print_tuple_impl!(@impl,);iter_print_tuple_impl!(@inc$C$c,,$($D$d)*);};(@inc$A:ident$a:ident,$($B:ident$b:ident)*,$C:ident$c:ident$($D:ident$d:ident)*)=>{iter_print_tuple_impl!(@impl$A$a,$($B$b)*);iter_print_tuple_impl!(@inc$A$a,$($B$b)*$C$c,$($D$d)*);};(@inc$A:ident$a:ident,$($B:ident$b:ident)*,)=>{iter_print_tuple_impl!(@impl$A$a,$($B$b)*);};($($t:tt)*)=>{iter_print_tuple_impl!(@inc,,$($t)*);};}iter_print_tuple_impl!(A a B b C c D d E e F f G g H h I i J j K k);#[doc=" Print expressions with a separator."]#[doc=" - `iter_print!(writer, args...)`"]#[doc=" - `@sep $expr`: set separator (default: `' '`)"]#[doc=" - `@ns`: alias for `@sep \"\"`"]#[doc=" - `@lf`: alias for `@sep '\\n'`"]#[doc=" - `@sp`: alias for `@sep ' '`"]#[doc=" - `@fmt $lit => {$($expr),*}`: print `format!($lit, $($expr),*)`"]#[doc=" - `@flush`: flush writer (auto insert `!`)"]#[doc=" - `@it $expr`: print iterator"]#[doc=" - `@it2d $expr`: print 2d-iterator"]#[doc=" - `@tup $expr`: print tuple (need to import [`IterPrint`])"]#[doc=" - `@ittup $expr`: print iterative tuple (need to import [`IterPrint`])"]#[doc=" - `$expr`: print expr"]#[doc=" - `{ args... }`: scoped"]#[doc=" - `;`: print `'\\n'`"]#[doc=" - `!`: not print `'\\n'` at the end"]#[macro_export]macro_rules!iter_print{(@@fmt$writer:expr,$sep:expr,$is_head:expr,$lit:literal,$($e:expr),*)=>{if!$is_head{::std::write!($writer,"{}",$sep).expect("io error");}::std::write!($writer,$lit,$($e),*).expect("io error");};(@@item$writer:expr,$sep:expr,$is_head:expr,$e:expr)=>{$crate::iter_print!(@@fmt$writer,$sep,$is_head,"{}",$e);};(@@line_feed$writer:expr$(,)?)=>{::std::writeln!($writer).expect("io error");};(@@it$writer:expr,$sep:expr,$is_head:expr,$iter:expr)=>{{let mut iter=$iter.into_iter();if let Some(item)=iter.next(){$crate::iter_print!(@@item$writer,$sep,$is_head,item);}for item in iter{$crate::iter_print!(@@item$writer,$sep,false,item);}}};(@@it2d$writer:expr,$sep:expr,$is_head:expr,$iter:expr)=>{let mut iter=$iter.into_iter();if let Some(item)=iter.next(){$crate::iter_print!(@@it$writer,$sep,$is_head,item);}for item in iter{$crate::iter_print!(@@line_feed$writer);$crate::iter_print!(@@it$writer,$sep,true,item);}};(@@tup$writer:expr,$sep:expr,$is_head:expr,$tuple:expr)=>{IterPrint::iter_print($tuple,&mut$writer,$sep,$is_head).expect("io error");};(@@ittup$writer:expr,$sep:expr,$is_head:expr,$iter:expr)=>{let mut iter=$iter.into_iter();if let Some(item)=iter.next(){$crate::iter_print!(@@tup$writer,$sep,$is_head,item);}for item in iter{$crate::iter_print!(@@line_feed$writer);$crate::iter_print!(@@tup$writer,$sep,true,item);}};(@@assert_tag item)=>{};(@@assert_tag it)=>{};(@@assert_tag it2d)=>{};(@@assert_tag tup)=>{};(@@assert_tag ittup)=>{};(@@assert_tag$tag:ident)=>{::std::compile_error!(::std::concat!("invalid tag in `iter_print!`: `",std::stringify!($tag),"`"));};(@@inner$writer:expr,$sep:expr,$is_head:expr,@sep$e:expr,$($t:tt)*)=>{$crate::iter_print!(@@inner$writer,$e,$is_head,$($t)*);};(@@inner$writer:expr,$sep:expr,$is_head:expr,@ns$($t:tt)*)=>{$crate::iter_print!(@@inner$writer,"",$is_head,$($t)*);};(@@inner$writer:expr,$sep:expr,$is_head:expr,@lf$($t:tt)*)=>{$crate::iter_print!(@@inner$writer,'\n',$is_head,$($t)*);};(@@inner$writer:expr,$sep:expr,$is_head:expr,@sp$($t:tt)*)=>{$crate::iter_print!(@@inner$writer,' ',$is_head,$($t)*);};(@@inner$writer:expr,$sep:expr,$is_head:expr,@flush$($t:tt)*)=>{$writer.flush().expect("io error");$crate::iter_print!(@@inner$writer,$sep,$is_head,!$($t)*);};(@@inner$writer:expr,$sep:expr,$is_head:expr,@fmt$lit:literal=>{$($e:expr),*$(,)?}$($t:tt)*)=>{$crate::iter_print!(@@fmt$writer,$sep,$is_head,$lit,$($e),*);$crate::iter_print!(@@inner$writer,$sep,$is_head,$($t)*);};(@@inner$writer:expr,$sep:expr,$is_head:expr,@$tag:ident$e:expr,$($t:tt)*)=>{$crate::iter_print!(@@assert_tag$tag);$crate::iter_print!(@@$tag$writer,$sep,$is_head,$e);$crate::iter_print!(@@inner$writer,$sep,false,$($t)*);};(@@inner$writer:expr,$sep:expr,$is_head:expr,@$tag:ident$e:expr;$($t:tt)*)=>{$crate::iter_print!(@@assert_tag$tag);$crate::iter_print!(@@$tag$writer,$sep,$is_head,$e);$crate::iter_print!(@@line_feed$writer);$crate::iter_print!(@@inner$writer,$sep,true,$($t)*);};(@@inner$writer:expr,$sep:expr,$is_head:expr,@$tag:ident$e:expr)=>{$crate::iter_print!(@@assert_tag$tag);$crate::iter_print!(@@$tag$writer,$sep,$is_head,$e);$crate::iter_print!(@@inner$writer,$sep,false,);};(@@inner$writer:expr,$sep:expr,$is_head:expr,@$tag:ident$($t:tt)*)=>{::std::compile_error!(::std::concat!("invalid expr in `iter_print!`: `",std::stringify!($($t)*),"`"));};(@@inner$writer:expr,$sep:expr,$is_head:expr,,$($t:tt)*)=>{$crate::iter_print!(@@inner$writer,$sep,$is_head,$($t)*);};(@@inner$writer:expr,$sep:expr,$is_head:expr,;$($t:tt)*)=>{$crate::iter_print!(@@line_feed$writer);$crate::iter_print!(@@inner$writer,$sep,$is_head,$($t)*);};(@@inner$writer:expr,$sep:expr,$is_head:expr,!$(,)?)=>{};(@@inner$writer:expr,$sep:expr,$is_head:expr,!$($t:tt)*)=>{$crate::iter_print!(@@inner$writer,$sep,$is_head,$($t)*);};(@@inner$writer:expr,$sep:expr,$is_head:expr,)=>{$crate::iter_print!(@@line_feed$writer);};(@@inner$writer:expr,$sep:expr,$is_head:expr,{$($t:tt)*}$($rest:tt)*)=>{$crate::iter_print!(@@inner$writer,$sep,$is_head,$($t)*,!);$crate::iter_print!(@@inner$writer,$sep,$is_head,$($rest)*);};(@@inner$writer:expr,$sep:expr,$is_head:expr,$($t:tt)*)=>{$crate::iter_print!(@@inner$writer,$sep,$is_head,@item$($t)*);};($writer:expr,$($t:tt)*)=>{{$crate::iter_print!(@@inner$writer,' ',true,$($t)*);}};}}
mod array{#[macro_export]macro_rules!array{[@inner$data:ident=[$init:expr;$len:expr]]=>{{use::std::mem::{ManuallyDrop,MaybeUninit};const LEN:usize=$len;let mut$data:[MaybeUninit<_>;LEN]=unsafe{MaybeUninit::uninit().assume_init()};$init;#[repr(C)]union __Transmuter<T:Copy>{src:ManuallyDrop<[MaybeUninit<T>;LEN]>,dst:ManuallyDrop<[T;LEN]>,}ManuallyDrop::into_inner(unsafe{__Transmuter{src:ManuallyDrop::new($data)}.dst})}};[||$e:expr;$len:expr]=>{$crate::array![@inner data=[data.iter_mut().for_each(|item|*item=MaybeUninit::new($e));$len]]};[|$i:pat|$e:expr;$len:expr]=>{$crate::array![@inner data=[data.iter_mut().enumerate().for_each(|($i,item)|*item=MaybeUninit::new($e));$len]]};[$e:expr;$len:expr]=>{{let e=$e;$crate::array![||Clone::clone(&e);$len]}};}}
pub use self::scanner::*;
mod scanner{use std::{iter::{from_fn,repeat_with,FromIterator},marker::PhantomData};pub fn read_stdin_all()->String{use std::io::Read as _;let mut s=String::new();std::io::stdin().read_to_string(&mut s).expect("io error");s}pub fn read_stdin_all_unchecked()->String{use std::io::Read as _;let mut buf=Vec::new();std::io::stdin().read_to_end(&mut buf).expect("io error");unsafe{String::from_utf8_unchecked(buf)}}pub fn read_all(mut reader:impl std::io::Read)->String{let mut s=String::new();reader.read_to_string(&mut s).expect("io error");s}pub fn read_all_unchecked(mut reader:impl std::io::Read)->String{let mut buf=Vec::new();reader.read_to_end(&mut buf).expect("io error");unsafe{String::from_utf8_unchecked(buf)}}pub fn read_stdin_line()->String{let mut s=String::new();std::io::stdin().read_line(&mut s).expect("io error");s}pub trait IterScan:Sized{type Output;fn scan<'a,I:Iterator<Item=&'a str>>(iter:&mut I)->Option<Self::Output>;}pub trait MarkedIterScan:Sized{type Output;fn mscan<'a,I:Iterator<Item=&'a str>>(self,iter:&mut I)->Option<Self::Output>;}#[derive(Clone,Debug)]pub struct Scanner<'a>{iter:std::str::SplitAsciiWhitespace<'a>}impl<'a>Scanner<'a>{#[inline]pub fn new(s:&'a str)->Self{let iter=s.split_ascii_whitespace();Self{iter}}#[inline]pub fn scan<T>(&mut self)-><T as IterScan>::Output where T:IterScan{<T as IterScan>::scan(&mut self.iter).expect("scan error")}#[inline]pub fn mscan<T>(&mut self,marker:T)-><T as MarkedIterScan>::Output where T:MarkedIterScan{marker.mscan(&mut self.iter).expect("scan error")}#[inline]pub fn scan_vec<T>(&mut self,size:usize)->Vec<<T as IterScan>::Output>where T:IterScan{(0..size).map(|_|<T as IterScan>::scan(&mut self.iter).expect("scan error")).collect()}#[inline]pub fn iter<'b,T>(&'b mut self)->ScannerIter<'a,'b,T>where T:IterScan{ScannerIter{inner:self,_marker:std::marker::PhantomData}}}macro_rules!iter_scan_impls{($($t:ty)*)=>{$(impl IterScan for$t{type Output=Self;#[inline]fn scan<'a,I:Iterator<Item=&'a str>>(iter:&mut I)->Option<Self>{iter.next()?.parse::<$t>().ok()}})*};}iter_scan_impls!(char u8 u16 u32 u64 usize i8 i16 i32 i64 isize f32 f64 u128 i128 String);macro_rules!iter_scan_tuple_impl{(@impl$($T:ident)*)=>{impl<$($T:IterScan),*>IterScan for($($T,)*){type Output=($(<$T as IterScan>::Output,)*);#[inline]fn scan<'a,It:Iterator<Item=&'a str>>(_iter:&mut It)->Option<Self::Output>{Some(($(<$T as IterScan>::scan(_iter)?,)*))}}};(@inner$($T:ident)*,)=>{iter_scan_tuple_impl!(@impl$($T)*);};(@inner$($T:ident)*,$U:ident$($Rest:ident)*)=>{iter_scan_tuple_impl!(@impl$($T)*);iter_scan_tuple_impl!(@inner$($T)*$U,$($Rest)*);};($($T:ident)*)=>{iter_scan_tuple_impl!(@inner,$($T)*);};}iter_scan_tuple_impl!(A B C D E F G H I J K);pub struct ScannerIter<'a,'b,T>{inner:&'b mut Scanner<'a>,_marker:std::marker::PhantomData<fn()->T>}impl<'a,'b,T>Iterator for ScannerIter<'a,'b,T>where T:IterScan{type Item=<T as IterScan>::Output;#[inline]fn next(&mut self)->Option<Self::Item>{<T as IterScan>::scan(&mut self.inner.iter)}}#[doc=" scan a value with Scanner"]#[doc=""]#[doc=" - `scan_value!(scanner, ELEMENT)`"]#[doc=""]#[doc=" ELEMENT :="]#[doc=" - `$ty`: IterScan"]#[doc=" - `@$expr`: MarkedIterScan"]#[doc=" - `[ELEMENT; $expr]`: vector"]#[doc=" - `[ELEMENT; const $expr]`: array"]#[doc=" - `[ELEMENT]`: iterator"]#[doc=" - `($(ELEMENT)*,)`: tuple"]#[macro_export]macro_rules!scan_value{(@repeat$scanner:expr,[$($t:tt)*]$($len:expr)?)=>{::std::iter::repeat_with(||$crate::scan_value!(@inner$scanner,[]$($t)*))$(.take($len).collect::<Vec<_>>())?};(@array$scanner:expr,[$($t:tt)*]$len:expr)=>{$crate::array![||$crate::scan_value!(@inner$scanner,[]$($t)*);$len]};(@tuple$scanner:expr,[$([$($args:tt)*])*])=>{($($($args)*,)*)};(@$tag:ident$scanner:expr,[[$($args:tt)*]])=>{$($args)*};(@$tag:ident$scanner:expr,[$($args:tt)*]@$e:expr)=>{$crate::scan_value!(@$tag$scanner,[$($args)*[$scanner.mscan($e)]])};(@$tag:ident$scanner:expr,[$($args:tt)*]@$e:expr,$($t:tt)*)=>{$crate::scan_value!(@$tag$scanner,[$($args)*[$scanner.mscan($e)]]$($t)*)};(@$tag:ident$scanner:expr,[$($args:tt)*]($($tuple:tt)*)$($t:tt)*)=>{$crate::scan_value!(@$tag$scanner,[$($args)*[$crate::scan_value!(@tuple$scanner,[]$($tuple)*)]]$($t)*)};(@$tag:ident$scanner:expr,[$($args:tt)*][@$e:expr;const$len:expr]$($t:tt)*)=>{$crate::scan_value!(@$tag$scanner,[$($args)*[$crate::scan_value!(@array$scanner,[@$e]$len)]]$($t)*)};(@$tag:ident$scanner:expr,[$($args:tt)*][@$e:expr;$len:expr]$($t:tt)*)=>{$crate::scan_value!(@$tag$scanner,[$($args)*[$crate::scan_value!(@repeat$scanner,[@$e]$len)]]$($t)*)};(@$tag:ident$scanner:expr,[$($args:tt)*][[$($tt:tt)*];const$len:expr]$($t:tt)*)=>{$crate::scan_value!(@$tag$scanner,[$($args)*[$crate::scan_value!(@array$scanner,[[$($tt)*]]$len)]]$($t)*)};(@$tag:ident$scanner:expr,[$($args:tt)*][[$($tt:tt)*];$len:expr]$($t:tt)*)=>{$crate::scan_value!(@$tag$scanner,[$($args)*[$crate::scan_value!(@repeat$scanner,[[$($tt)*]]$len)]]$($t)*)};(@$tag:ident$scanner:expr,[$($args:tt)*][($($tt:tt)*);const$len:expr]$($t:tt)*)=>{$crate::scan_value!(@$tag$scanner,[$($args)*[$crate::scan_value!(@array$scanner,[($($tt)*)]$len)]]$($t)*)};(@$tag:ident$scanner:expr,[$($args:tt)*][($($tt:tt)*);$len:expr]$($t:tt)*)=>{$crate::scan_value!(@$tag$scanner,[$($args)*[$crate::scan_value!(@repeat$scanner,[($($tt)*)]$len)]]$($t)*)};(@$tag:ident$scanner:expr,[$($args:tt)*][$ty:ty;const$len:expr]$($t:tt)*)=>{$crate::scan_value!(@$tag$scanner,[$($args)*[$crate::scan_value!(@array$scanner,[$ty]$len)]]$($t)*)};(@$tag:ident$scanner:expr,[$($args:tt)*][$ty:ty;$len:expr]$($t:tt)*)=>{$crate::scan_value!(@$tag$scanner,[$($args)*[$crate::scan_value!(@repeat$scanner,[$ty]$len)]]$($t)*)};(@$tag:ident$scanner:expr,[$($args:tt)*][$($tt:tt)*]$($t:tt)*)=>{$crate::scan_value!(@$tag$scanner,[$($args)*[$crate::scan_value!(@repeat$scanner,[$($tt)*])]]$($t)*)};(@$tag:ident$scanner:expr,[$($args:tt)*]$ty:ty)=>{$crate::scan_value!(@$tag$scanner,[$($args)*[$scanner.scan::<$ty>()]])};(@$tag:ident$scanner:expr,[$($args:tt)*]$ty:ty,$($t:tt)*)=>{$crate::scan_value!(@$tag$scanner,[$($args)*[$scanner.scan::<$ty>()]]$($t)*)};(@$tag:ident$scanner:expr,[$($args:tt)*],$($t:tt)*)=>{$crate::scan_value!(@$tag$scanner,[$($args)*]$($t)*)};(@$tag:ident$scanner:expr,[$($args:tt)*])=>{::std::compile_error!(::std::stringify!($($args)*))};($scanner:expr,$($t:tt)*)=>{$crate::scan_value!(@inner$scanner,[]$($t)*)}}#[doc=" scan and bind values with Scanner"]#[doc=""]#[doc=" - `scan!(scanner, $($pat $(: ELEMENT)?),*)`"]#[macro_export]macro_rules!scan{(@assert$p:pat)=>{};(@assert$($p:tt)*)=>{::std::compile_error!(::std::concat!("expected pattern, found `",::std::stringify!($($p)*),"`"));};(@pat$scanner:expr,[][])=>{};(@pat$scanner:expr,[][],$($t:tt)*)=>{$crate::scan!(@pat$scanner,[][]$($t)*)};(@pat$scanner:expr,[$($p:tt)*][]$x:ident$($t:tt)*)=>{$crate::scan!(@pat$scanner,[$($p)*$x][]$($t)*)};(@pat$scanner:expr,[$($p:tt)*][]::$($t:tt)*)=>{$crate::scan!(@pat$scanner,[$($p)*::][]$($t)*)};(@pat$scanner:expr,[$($p:tt)*][]&$($t:tt)*)=>{$crate::scan!(@pat$scanner,[$($p)*&][]$($t)*)};(@pat$scanner:expr,[$($p:tt)*][]($($x:tt)*)$($t:tt)*)=>{$crate::scan!(@pat$scanner,[$($p)*($($x)*)][]$($t)*)};(@pat$scanner:expr,[$($p:tt)*][][$($x:tt)*]$($t:tt)*)=>{$crate::scan!(@pat$scanner,[$($p)*[$($x)*]][]$($t)*)};(@pat$scanner:expr,[$($p:tt)*][]{$($x:tt)*}$($t:tt)*)=>{$crate::scan!(@pat$scanner,[$($p)*{$($x)*}][]$($t)*)};(@pat$scanner:expr,[$($p:tt)*][]:$($t:tt)*)=>{$crate::scan!(@ty$scanner,[$($p)*][]$($t)*)};(@pat$scanner:expr,[$($p:tt)*][]$($t:tt)*)=>{$crate::scan!(@let$scanner,[$($p)*][usize]$($t)*)};(@ty$scanner:expr,[$($p:tt)*][$($tt:tt)*]@$e:expr)=>{$crate::scan!(@let$scanner,[$($p)*][$($tt)*@$e])};(@ty$scanner:expr,[$($p:tt)*][$($tt:tt)*]@$e:expr,$($t:tt)*)=>{$crate::scan!(@let$scanner,[$($p)*][$($tt)*@$e],$($t)*)};(@ty$scanner:expr,[$($p:tt)*][$($tt:tt)*]($($x:tt)*)$($t:tt)*)=>{$crate::scan!(@let$scanner,[$($p)*][$($tt)*($($x)*)]$($t)*)};(@ty$scanner:expr,[$($p:tt)*][$($tt:tt)*][$($x:tt)*]$($t:tt)*)=>{$crate::scan!(@let$scanner,[$($p)*][$($tt)*[$($x)*]]$($t)*)};(@ty$scanner:expr,[$($p:tt)*][$($tt:tt)*]$ty:ty)=>{$crate::scan!(@let$scanner,[$($p)*][$($tt)*$ty])};(@ty$scanner:expr,[$($p:tt)*][$($tt:tt)*]$ty:ty,$($t:tt)*)=>{$crate::scan!(@let$scanner,[$($p)*][$($tt)*$ty],$($t)*)};(@let$scanner:expr,[$($p:tt)*][$($tt:tt)*]$($t:tt)*)=>{$crate::scan!{@assert$($p)*}let$($p)* =$crate::scan_value!($scanner,$($tt)*);$crate::scan!(@pat$scanner,[][]$($t)*)};($scanner:expr,$($t:tt)*)=>{$crate::scan!(@pat$scanner,[][]$($t)*)}}#[derive(Debug,Copy,Clone)]pub enum Usize1{}impl IterScan for Usize1{type Output=usize;#[inline]fn scan<'a,I:Iterator<Item=&'a str>>(iter:&mut I)->Option<Self::Output>{<usize as IterScan>::scan(iter)?.checked_sub(1)}}#[derive(Debug,Copy,Clone)]pub struct CharWithBase(pub char);impl MarkedIterScan for CharWithBase{type Output=usize;#[inline]fn mscan<'a,I:Iterator<Item=&'a str>>(self,iter:&mut I)->Option<Self::Output>{Some((<char as IterScan>::scan(iter)?as u8-self.0 as u8)as usize)}}#[derive(Debug,Copy,Clone)]pub enum Chars{}impl IterScan for Chars{type Output=Vec<char>;#[inline]fn scan<'a,I:Iterator<Item=&'a str>>(iter:&mut I)->Option<Self::Output>{Some(iter.next()?.chars().collect())}}#[derive(Debug,Copy,Clone)]pub struct CharsWithBase(pub char);impl MarkedIterScan for CharsWithBase{type Output=Vec<usize>;#[inline]fn mscan<'a,I:Iterator<Item=&'a str>>(self,iter:&mut I)->Option<Self::Output>{Some(iter.next()?.chars().map(|c|(c as u8-self.0 as u8)as usize).collect())}}#[derive(Debug,Copy,Clone)]pub enum Byte1{}impl IterScan for Byte1{type Output=u8;#[inline]fn scan<'a,I:Iterator<Item=&'a str>>(iter:&mut I)->Option<Self::Output>{let bytes=iter.next()?.as_bytes();assert_eq!(bytes.len(),1);Some(bytes[0])}}#[derive(Debug,Copy,Clone)]pub struct ByteWithBase(pub u8);impl MarkedIterScan for ByteWithBase{type Output=usize;#[inline]fn mscan<'a,I:Iterator<Item=&'a str>>(self,iter:&mut I)->Option<Self::Output>{Some((<char as IterScan>::scan(iter)?as u8-self.0)as usize)}}#[derive(Debug,Copy,Clone)]pub enum Bytes{}impl IterScan for Bytes{type Output=Vec<u8>;#[inline]fn scan<'a,I:Iterator<Item=&'a str>>(iter:&mut I)->Option<Self::Output>{Some(iter.next()?.bytes().collect())}}#[derive(Debug,Copy,Clone)]pub struct BytesWithBase(pub u8);impl MarkedIterScan for BytesWithBase{type Output=Vec<usize>;#[inline]fn mscan<'a,I:Iterator<Item=&'a str>>(self,iter:&mut I)->Option<Self::Output>{Some(iter.next()?.bytes().map(|c|(c-self.0)as usize).collect())}}#[derive(Debug,Copy,Clone)]pub struct Collect<T,B=Vec<<T as IterScan>::Output>>where T:IterScan,B:FromIterator<<T as IterScan>::Output>{size:usize,_marker:PhantomData<fn()->(T,B)>}impl<T,B>Collect<T,B>where T:IterScan,B:FromIterator<<T as IterScan>::Output>{pub fn new(size:usize)->Self{Self{size,_marker:PhantomData}}}impl<T,B>MarkedIterScan for Collect<T,B>where T:IterScan,B:FromIterator<<T as IterScan>::Output>{type Output=B;#[inline]fn mscan<'a,I:Iterator<Item=&'a str>>(self,iter:&mut I)->Option<Self::Output>{repeat_with(||<T as IterScan>::scan(iter)).take(self.size).collect()}}#[derive(Debug,Copy,Clone)]pub struct SizedCollect<T,B=Vec<<T as IterScan>::Output>>where T:IterScan,B:FromIterator<<T as IterScan>::Output>{_marker:PhantomData<fn()->(T,B)>}impl<T,B>IterScan for SizedCollect<T,B>where T:IterScan,B:FromIterator<<T as IterScan>::Output>{type Output=B;#[inline]fn scan<'a,I:Iterator<Item=&'a str>>(iter:&mut I)->Option<Self::Output>{let size=usize::scan(iter)?;repeat_with(||<T as IterScan>::scan(iter)).take(size).collect()}}#[derive(Debug,Copy,Clone)]pub struct Splitted<T,P>where T:IterScan{pat:P,_marker:PhantomData<fn()->T>}impl<T,P>Splitted<T,P>where T:IterScan{pub fn new(pat:P)->Self{Self{pat,_marker:PhantomData}}}impl<T>MarkedIterScan for Splitted<T,char>where T:IterScan{type Output=Vec<<T as IterScan>::Output>;fn mscan<'a,I:Iterator<Item=&'a str>>(self,iter:&mut I)->Option<Self::Output>{let mut iter=iter.next()?.split(self.pat);Some(from_fn(||<T as IterScan>::scan(&mut iter)).collect())}}impl<T>MarkedIterScan for Splitted<T,&str>where T:IterScan{type Output=Vec<<T as IterScan>::Output>;fn mscan<'a,I:Iterator<Item=&'a str>>(self,iter:&mut I)->Option<Self::Output>{let mut iter=iter.next()?.split(self.pat);Some(from_fn(||<T as IterScan>::scan(&mut iter)).collect())}}impl<T,F>MarkedIterScan for F where F:Fn(&str)->Option<T>{type Output=T;fn mscan<'a,I:Iterator<Item=&'a str>>(self,iter:&mut I)->Option<Self::Output>{self(iter.next()?)}}}
pub use self::number_theoretic_transform::{Convolve,Convolve998244353,MIntConvolve};
mod number_theoretic_transform{use super::{montgomery::*,AssociatedValue,ConvolveSteps,MInt,MIntBase,MIntConvert,One,Zero};use std::marker::PhantomData;pub trait NttModulus:'static+Sized+MIntBase<Inner=u32>+MIntConvert<u32>+AssociatedValue<T=NttCache<Self>>{fn primitive_root()->MInt<Self>;}pub struct Convolve<M>(PhantomData<fn()->M>);pub type Convolve998244353=Convolve<Modulo998244353>;pub type MIntConvolve<M>=Convolve<(M,(Modulo2013265921,Modulo1811939329,Modulo2113929217))>;macro_rules!impl_ntt_modulus{($([$name:ident,$g:expr]),*)=>{$(impl NttModulus for$name{fn primitive_root()->MInt<Self>{MInt::new_unchecked($g)}}crate::impl_assoc_value!($name,NttCache<$name>,NttCache::new());)*};}impl_ntt_modulus!([Modulo998244353,3],[Modulo2113929217,5],[Modulo1811939329,13],[Modulo2013265921,31]);#[derive(Debug)]pub struct NttCache<M>where M:NttModulus{cache:Vec<MInt<M>>,icache:Vec<MInt<M>>}impl<M>Clone for NttCache<M>where M:NttModulus{fn clone(&self)->Self{Self{cache:self.cache.clone(),icache:self.icache.clone()}}}impl<M>NttCache<M>where M:NttModulus{fn new()->Self{Self{cache:Vec::new(),icache:Vec::new()}}fn ensure(&mut self,n:usize){assert_eq!(n.count_ones(),1,"call with power of two but {}",n);let mut m=self.cache.len();assert!(m.count_ones()<=1,"length might be power of two but {}",m);if m>=n{return;}let q:usize=M::mod_into()as usize-1;self.cache.reserve_exact(n-m);self.icache.reserve_exact(n-m);if self.cache.is_empty(){self.cache.push(MInt::one());self.icache.push(MInt::one());m+=1;}while m<n{let p=M::primitive_root().pow(q/(m*4));let pinv=p.inv();for i in 0..m{self.cache.push(self.cache[i]*p);self.icache.push(self.icache[i]*pinv);}m<<=1;}assert_eq!(self.cache.len(),n);}}impl<M>NttCache<M>where M:NttModulus{fn ntt(a:&mut[MInt<M>]){M::modify(|cache|{let n=a.len();cache.ensure(n/2);let mut v=n/2;while v>0{for(a,wj)in a.chunks_exact_mut(v<<1).zip(&cache.cache){let(l,r)=a.split_at_mut(v);for(x,y)in l.iter_mut().zip(r){let ajv=wj**y;*y=*x-ajv;*x+=ajv;}}v>>=1;}});}fn intt(a:&mut[MInt<M>]){M::modify(|cache|{let n=a.len();cache.ensure(n/2);let mut v=1;while v<n{for(a,wj)in a.chunks_exact_mut(v<<1).zip(&cache.icache){let(l,r)=a.split_at_mut(v);for(x,y)in l.iter_mut().zip(r){let ajv=*x-*y;*x+=*y;*y=wj*ajv;}}v<<=1;}});}}impl<M>ConvolveSteps for Convolve<M>where M:NttModulus{type T=Vec<MInt<M>>;type F=Vec<MInt<M>>;fn length(t:&Self::T)->usize{t.len()}fn transform(mut t:Self::T,len:usize)->Self::F{t.resize_with(len.max(2).next_power_of_two(),Zero::zero);NttCache::<M>::ntt(&mut t);t}fn inverse_transform(mut f:Self::F,len:usize)->Self::T{NttCache::<M>::intt(&mut f);f.truncate(len);let inv=MInt::from(len.max(2).next_power_of_two()as u32).inv();for f in f.iter_mut(){*f*=inv;}f}fn multiply(f:&mut Self::F,g:&Self::F){assert_eq!(f.len(),g.len());for(f,g)in f.iter_mut().zip(g.iter()){*f*=*g;}}}type MVec<M>=Vec<MInt<M>>;impl<M,N1,N2,N3>ConvolveSteps for Convolve<(M,(N1,N2,N3))>where M:MIntConvert+MIntConvert<u32>,N1:NttModulus,N2:NttModulus,N3:NttModulus{type T=MVec<M>;type F=(MVec<N1>,MVec<N2>,MVec<N3>);fn length(t:&Self::T)->usize{t.len()}fn transform(t:Self::T,len:usize)->Self::F{let npot=len.max(2).next_power_of_two();let mut f=(MVec::<N1>::with_capacity(npot),MVec::<N2>::with_capacity(npot),MVec::<N3>::with_capacity(npot));for t in t{f.0.push(<M as MIntConvert<u32>>::into(t.inner()).into());f.1.push(<M as MIntConvert<u32>>::into(t.inner()).into());f.2.push(<M as MIntConvert<u32>>::into(t.inner()).into());}f.0.resize_with(npot,Zero::zero);f.1.resize_with(npot,Zero::zero);f.2.resize_with(npot,Zero::zero);NttCache::<N1>::ntt(&mut f.0);NttCache::<N2>::ntt(&mut f.1);NttCache::<N3>::ntt(&mut f.2);f}fn inverse_transform(f:Self::F,len:usize)->Self::T{let t1=MInt::<N2>::new(N1::get_mod()).inv();let m1=MInt::<M>::from(N1::get_mod());let m1_3=MInt::<N3>::new(N1::get_mod());let t2=(m1_3*MInt::<N3>::new(N2::get_mod())).inv();let m2=m1*MInt::<M>::from(N2::get_mod());Convolve::<N1>::inverse_transform(f.0,len).into_iter().zip(Convolve::<N2>::inverse_transform(f.1,len)).zip(Convolve::<N3>::inverse_transform(f.2,len)).map(|((c1,c2),c3)|{let d1=c1.inner();let d2=((c2-MInt::<N2>::from(d1))*t1).inner();let x=MInt::<N3>::new(d1)+MInt::<N3>::new(d2)*m1_3;let d3=((c3-x)*t2).inner();MInt::<M>::from(d1)+MInt::<M>::from(d2)*m1+MInt::<M>::from(d3)*m2}).collect()}fn multiply(f:&mut Self::F,g:&Self::F){assert_eq!(f.0.len(),g.0.len());assert_eq!(f.1.len(),g.1.len());assert_eq!(f.2.len(),g.2.len());for(f,g)in f.0.iter_mut().zip(g.0.iter()){*f*=*g;}for(f,g)in f.1.iter_mut().zip(g.1.iter()){*f*=*g;}for(f,g)in f.2.iter_mut().zip(g.2.iter()){*f*=*g;}}}}
pub use self::associated_value::AssociatedValue;
mod associated_value{#[doc=" Trait for a modifiable value associated with a type."]pub trait AssociatedValue{#[doc=" Type of value."]type T:'static+Clone;unsafe fn __local_key()->&'static std::thread::LocalKey<std::cell::Cell<Self::T>>;#[inline]fn get()->Self::T{Self::with(Clone::clone)}#[inline]fn set(x:Self::T){unsafe{Self::__local_key().with(|cell|cell.set(x))}}#[inline]fn replace(x:Self::T)->Self::T{unsafe{Self::__local_key().with(|cell|cell.replace(x))}}#[inline]fn with<F,R>(f:F)->R where F:FnOnce(&Self::T)->R{unsafe{Self::__local_key().with(|cell|f(&*cell.as_ptr()))}}#[inline]fn modify<F,R>(f:F)->R where F:FnOnce(&mut Self::T)->R{unsafe{Self::__local_key().with(|cell|f(&mut*cell.as_ptr()))}}}#[doc=" Implement [`AssociatedValue`]."]#[doc=""]#[doc=" [`AssociatedValue`]: super::AssociatedValue"]#[doc=""]#[doc=" # Examples"]#[doc=""]#[doc=" ```"]#[doc=" use competitive::tools::AssociatedValue;"]#[doc=" enum X {}"]#[doc=" competitive::impl_assoc_value!(X, usize, 1);"]#[doc=" assert_eq!(X::get(), 1);"]#[doc=" X::set(10);"]#[doc=" assert_eq!(X::get(), 10);"]#[doc=" ```"]#[doc=""]#[doc=" init with `Default::default()`"]#[doc=""]#[doc=" ```"]#[doc=" use competitive::tools::AssociatedValue;"]#[doc=" enum X {}"]#[doc=" competitive::impl_assoc_value!(X, usize);"]#[doc=" assert_eq!(X::get(), Default::default());"]#[doc=" ```"]#[macro_export]macro_rules!impl_assoc_value{($name:ident,$t:ty)=>{$crate::impl_assoc_value!($name,$t,Default::default());};($name:ident,$t:ty,$e:expr)=>{impl AssociatedValue for$name{type T=$t;#[inline]unsafe fn __local_key()->&'static::std::thread::LocalKey<::std::cell::Cell<Self::T>>{::std::thread_local!(static __LOCAL_KEY: ::std::cell::Cell<$t> =::std::cell::Cell::new($e));&__LOCAL_KEY}}};}}
pub use self::convolve_steps::ConvolveSteps;
mod convolve_steps{pub trait ConvolveSteps{type T;type F;fn length(t:&Self::T)->usize;fn transform(t:Self::T,len:usize)->Self::F;fn inverse_transform(f:Self::F,len:usize)->Self::T;fn multiply(f:&mut Self::F,g:&Self::F);fn convolve(a:Self::T,b:Self::T)->Self::T{let len=(Self::length(&a)+Self::length(&b)).saturating_sub(1);let mut a=Self::transform(a,len);let b=Self::transform(b,len);Self::multiply(&mut a,&b);Self::inverse_transform(a,len)}}}
pub use mint_base::{MInt,MIntBase,MIntConvert};
mod mint_base{use super::*;use std::{fmt::{self,Debug,Display},hash::{Hash,Hasher},iter::{Product,Sum},marker::PhantomData,ops::{Add,AddAssign,Div,DivAssign,Mul,MulAssign,Neg,Sub,SubAssign},str::FromStr};#[repr(transparent)]pub struct MInt<M>where M:MIntBase{x:M::Inner,_marker:PhantomData<fn()->M>}pub trait MIntConvert<T=<Self as MIntBase>::Inner>:MIntBase{fn from(x:T)-><Self as MIntBase>::Inner;fn into(x:<Self as MIntBase>::Inner)->T;fn mod_into()->T;}pub trait MIntBase{type Inner:Sized+Copy+Eq+Debug+Hash;fn get_mod()->Self::Inner;fn mod_zero()->Self::Inner;fn mod_one()->Self::Inner;fn mod_add(x:Self::Inner,y:Self::Inner)->Self::Inner;fn mod_sub(x:Self::Inner,y:Self::Inner)->Self::Inner;fn mod_mul(x:Self::Inner,y:Self::Inner)->Self::Inner;fn mod_div(x:Self::Inner,y:Self::Inner)->Self::Inner;fn mod_neg(x:Self::Inner)->Self::Inner;fn mod_inv(x:Self::Inner)->Self::Inner;fn mod_pow(x:Self::Inner,y:usize)->Self::Inner{let(mut x,mut y,mut z)=(x,y,Self::mod_one());while y>0{if y&1==1{z=Self::mod_mul(z,x);}x=Self::mod_mul(x,x);y>>=1;}z}}impl<M>MInt<M>where M:MIntConvert{#[inline]pub fn new(x:M::Inner)->Self{Self::new_unchecked(<M as MIntConvert<M::Inner>>::from(x))}#[inline]pub fn inner(self)->M::Inner{<M as MIntConvert<M::Inner>>::into(self.x)}}impl<M>MInt<M>where M:MIntBase{#[inline]pub fn new_unchecked(x:M::Inner)->Self{Self{x,_marker:PhantomData}}#[inline]pub fn get_mod()->M::Inner{M::get_mod()}#[inline]pub fn pow(self,y:usize)->Self{Self::new_unchecked(M::mod_pow(self.x,y))}#[inline]pub fn inv(self)->Self{Self::new_unchecked(M::mod_inv(self.x))}}impl<M>Clone for MInt<M>where M:MIntBase{#[inline]fn clone(&self)->Self{Self{x:Clone::clone(&self.x),_marker:PhantomData}}}impl<M>Copy for MInt<M>where M:MIntBase{}impl<M>Debug for MInt<M>where M:MIntBase{fn fmt(&self,f:&mut fmt::Formatter<'_>)->fmt::Result{Debug::fmt(&self.x,f)}}impl<M>Default for MInt<M>where M:MIntBase{#[inline]fn default()->Self{<Self as Zero>::zero()}}impl<M>PartialEq for MInt<M>where M:MIntBase{#[inline]fn eq(&self,other:&Self)->bool{PartialEq::eq(&self.x,&other.x)}}impl<M>Eq for MInt<M>where M:MIntBase{}impl<M>Hash for MInt<M>where M:MIntBase{#[inline]fn hash<H:Hasher>(&self,state:&mut H){Hash::hash(&self.x,state)}}macro_rules!impl_mint_from{($($t:ty),*)=>{$(impl<M>From<$t>for MInt<M>where M:MIntConvert<$t>,{#[inline]fn from(x:$t)->Self{Self::new_unchecked(<M as MIntConvert<$t>>::from(x))}}impl<M>From<MInt<M>>for$t where M:MIntConvert<$t>,{#[inline]fn from(x:MInt<M>)->$t{<M as MIntConvert<$t>>::into(x.x)}})*};}impl_mint_from!(u8,u16,u32,u64,u128,usize,i8,i16,i32,i64,i128,isize);impl<M>Zero for MInt<M>where M:MIntBase{#[inline]fn zero()->Self{Self::new_unchecked(M::mod_zero())}}impl<M>One for MInt<M>where M:MIntBase{#[inline]fn one()->Self{Self::new_unchecked(M::mod_one())}}impl<M>Add for MInt<M>where M:MIntBase{type Output=Self;#[inline]fn add(self,rhs:Self)->Self::Output{Self::new_unchecked(M::mod_add(self.x,rhs.x))}}impl<M>Sub for MInt<M>where M:MIntBase{type Output=Self;#[inline]fn sub(self,rhs:Self)->Self::Output{Self::new_unchecked(M::mod_sub(self.x,rhs.x))}}impl<M>Mul for MInt<M>where M:MIntBase{type Output=Self;#[inline]fn mul(self,rhs:Self)->Self::Output{Self::new_unchecked(M::mod_mul(self.x,rhs.x))}}impl<M>Div for MInt<M>where M:MIntBase{type Output=Self;#[inline]fn div(self,rhs:Self)->Self::Output{Self::new_unchecked(M::mod_div(self.x,rhs.x))}}impl<M>Neg for MInt<M>where M:MIntBase{type Output=Self;#[inline]fn neg(self)->Self::Output{Self::new_unchecked(M::mod_neg(self.x))}}impl<M>Sum for MInt<M>where M:MIntBase{#[inline]fn sum<I:Iterator<Item=Self>>(iter:I)->Self{iter.fold(<Self as Zero>::zero(),Add::add)}}impl<M>Product for MInt<M>where M:MIntBase{#[inline]fn product<I:Iterator<Item=Self>>(iter:I)->Self{iter.fold(<Self as One>::one(),Mul::mul)}}impl<'a,M:'a>Sum<&'a MInt<M>>for MInt<M>where M:MIntBase{#[inline]fn sum<I:Iterator<Item=&'a Self>>(iter:I)->Self{iter.fold(<Self as Zero>::zero(),Add::add)}}impl<'a,M:'a>Product<&'a MInt<M>>for MInt<M>where M:MIntBase{#[inline]fn product<I:Iterator<Item=&'a Self>>(iter:I)->Self{iter.fold(<Self as One>::one(),Mul::mul)}}impl<M>Display for MInt<M>where M:MIntConvert,M::Inner:Display{fn fmt<'a>(&self,f:&mut fmt::Formatter<'a>)->Result<(),fmt::Error>{write!(f,"{}",self.inner())}}impl<M>FromStr for MInt<M>where M:MIntConvert,M::Inner:FromStr{type Err=<M::Inner as FromStr>::Err;#[inline]fn from_str(s:&str)->Result<Self,Self::Err>{s.parse::<M::Inner>().map(Self::new)}}impl<M>IterScan for MInt<M>where M:MIntConvert,M::Inner:FromStr{type Output=Self;#[inline]fn scan<'a,I:Iterator<Item=&'a str>>(iter:&mut I)->Option<Self::Output>{iter.next()?.parse::<MInt<M>>().ok()}}macro_rules!impl_mint_ref_binop{($imp:ident,$method:ident,$t:ty)=>{impl<M>$imp<$t>for&$t where M:MIntBase,{type Output=<$t as$imp<$t>>::Output;#[inline]fn$method(self,other:$t)-><$t as$imp<$t>>::Output{$imp::$method(*self,other)}}impl<M>$imp<&$t>for$t where M:MIntBase,{type Output=<$t as$imp<$t>>::Output;#[inline]fn$method(self,other:&$t)-><$t as$imp<$t>>::Output{$imp::$method(self,*other)}}impl<M>$imp<&$t>for&$t where M:MIntBase,{type Output=<$t as$imp<$t>>::Output;#[inline]fn$method(self,other:&$t)-><$t as$imp<$t>>::Output{$imp::$method(*self,*other)}}};}impl_mint_ref_binop!(Add,add,MInt<M>);impl_mint_ref_binop!(Sub,sub,MInt<M>);impl_mint_ref_binop!(Mul,mul,MInt<M>);impl_mint_ref_binop!(Div,div,MInt<M>);macro_rules!impl_mint_ref_unop{($imp:ident,$method:ident,$t:ty)=>{impl<M>$imp for&$t where M:MIntBase,{type Output=<$t as$imp>::Output;#[inline]fn$method(self)-><$t as$imp>::Output{$imp::$method(*self)}}};}impl_mint_ref_unop!(Neg,neg,MInt<M>);macro_rules!impl_mint_ref_op_assign{($imp:ident,$method:ident,$t:ty,$fromimp:ident,$frommethod:ident)=>{impl<M>$imp<$t>for$t where M:MIntBase,{#[inline]fn$method(&mut self,rhs:$t){*self=$fromimp::$frommethod(*self,rhs);}}impl<M>$imp<&$t>for$t where M:MIntBase,{#[inline]fn$method(&mut self,other:&$t){$imp::$method(self,*other);}}};}impl_mint_ref_op_assign!(AddAssign,add_assign,MInt<M>,Add,add);impl_mint_ref_op_assign!(SubAssign,sub_assign,MInt<M>,Sub,sub);impl_mint_ref_op_assign!(MulAssign,mul_assign,MInt<M>,Mul,mul);impl_mint_ref_op_assign!(DivAssign,div_assign,MInt<M>,Div,div);}
pub mod montgomery{use super::*;impl<M>MIntBase for M where M:MontgomeryReduction32{type Inner=u32;#[inline]fn get_mod()->Self::Inner{<Self as MontgomeryReduction32>::get_mod()}#[inline]fn mod_zero()->Self::Inner{0}#[inline]fn mod_one()->Self::Inner{Self::n1()}#[inline]fn mod_add(x:Self::Inner,y:Self::Inner)->Self::Inner{let z=x+y;let m=Self::get_mod();if z>=m{z-m}else{z}}#[inline]fn mod_sub(x:Self::Inner,y:Self::Inner)->Self::Inner{if x<y{x+Self::get_mod()-y}else{x-y}}#[inline]fn mod_mul(x:Self::Inner,y:Self::Inner)->Self::Inner{Self::reduce(x as u64*y as u64)}#[inline]fn mod_div(x:Self::Inner,y:Self::Inner)->Self::Inner{Self::mod_mul(x,Self::mod_inv(y))}#[inline]fn mod_neg(x:Self::Inner)->Self::Inner{if x==0{0}else{Self::get_mod()-x}}fn mod_inv(x:Self::Inner)->Self::Inner{let p=Self::get_mod()as i32;let(mut a,mut b)=(x as i32,p);let(mut u,mut x)=(1,0);while a!=0{let k=b/a;x-=k*u;b-=k*a;std::mem::swap(&mut x,&mut u);std::mem::swap(&mut b,&mut a);}Self::reduce((if x<0{x+p}else{x})as u64*Self::n3()as u64)}}impl<M>MIntConvert<u32>for M where M:MontgomeryReduction32{#[inline]fn from(x:u32)->Self::Inner{Self::reduce(x as u64*Self::n2()as u64)}#[inline]fn into(x:Self::Inner)->u32{Self::reduce(x as u64)}#[inline]fn mod_into()->u32{<Self as MIntBase>::get_mod()}}impl<M>MIntConvert<u64>for M where M:MontgomeryReduction32{#[inline]fn from(x:u64)->Self::Inner{Self::reduce(x%Self::get_mod()as u64*Self::n2()as u64)}#[inline]fn into(x:Self::Inner)->u64{Self::reduce(x as u64)as u64}#[inline]fn mod_into()->u64{<Self as MIntBase>::get_mod()as u64}}impl<M>MIntConvert<usize>for M where M:MontgomeryReduction32{#[inline]fn from(x:usize)->Self::Inner{Self::reduce(x as u64%Self::get_mod()as u64*Self::n2()as u64)}#[inline]fn into(x:Self::Inner)->usize{Self::reduce(x as u64)as usize}#[inline]fn mod_into()->usize{<Self as MIntBase>::get_mod()as usize}}impl<M>MIntConvert<i32>for M where M:MontgomeryReduction32{#[inline]fn from(x:i32)->Self::Inner{let x=x%<Self as MIntBase>::get_mod()as i32;let x=if x<0{(x+<Self as MIntBase>::get_mod()as i32)as u64}else{x as u64};Self::reduce(x*Self::n2()as u64)}#[inline]fn into(x:Self::Inner)->i32{Self::reduce(x as u64)as i32}#[inline]fn mod_into()->i32{<Self as MIntBase>::get_mod()as i32}}impl<M>MIntConvert<i64>for M where M:MontgomeryReduction32{#[inline]fn from(x:i64)->Self::Inner{let x=x%<Self as MIntBase>::get_mod()as i64;let x=if x<0{(x+<Self as MIntBase>::get_mod()as i64)as u64}else{x as u64};Self::reduce(x*Self::n2()as u64)}#[inline]fn into(x:Self::Inner)->i64{Self::reduce(x as u64)as i64}#[inline]fn mod_into()->i64{<Self as MIntBase>::get_mod()as i64}}impl<M>MIntConvert<isize>for M where M:MontgomeryReduction32{#[inline]fn from(x:isize)->Self::Inner{let x=x%<Self as MIntBase>::get_mod()as isize;let x=if x<0{(x+<Self as MIntBase>::get_mod()as isize)as u64}else{x as u64};Self::reduce(x*Self::n2()as u64)}#[inline]fn into(x:Self::Inner)->isize{Self::reduce(x as u64)as isize}#[inline]fn mod_into()->isize{<Self as MIntBase>::get_mod()as isize}}#[doc=" m is prime, n = 2^32"]pub trait MontgomeryReduction32{#[doc=" m"]fn get_mod()->u32;#[doc=" (-m)^{-1} mod n"]fn r()->u32{let m=Self::get_mod();let mut r=0;let mut t=0;for i in 0..32{if t%2==0{t+=m;r+=1<<i;}t/=2;}r}#[doc=" n^1 mod m"]fn n1()->u32;#[doc=" n^2 mod m"]fn n2()->u32;#[doc=" n^3 mod m"]fn n3()->u32;#[doc=" n^{-1}x = (x + (xr mod n)m) / n"]fn reduce(x:u64)->u32{let m:u32=Self::get_mod();let r=Self::r();let mut x=((x+r.wrapping_mul(x as u32)as u64*m as u64)>>32)as u32;if x>=m{x-=m;}x}}macro_rules!define_montgomery_reduction_32{($([$name:ident,$m:expr,$r:expr,$n1:expr,$n2:expr,$n3:expr,$mint_name:ident$(,)?]),*$(,)?)=>{$(pub enum$name{}impl MontgomeryReduction32 for$name{#[inline]fn get_mod()->u32{$m}#[inline]fn r()->u32{$r}#[inline]fn n1()->u32{$n1}#[inline]fn n2()->u32{$n2}#[inline]fn n3()->u32{$n3}}pub type$mint_name=MInt<$name>;)*};}define_montgomery_reduction_32!([Modulo998244353,998_244_353,998_244_351,301_989_884,932_051_910,679_058_953,MInt998244353,],[Modulo2113929217,2_113_929_217,2_113_929_215,67_108_862,2_111_798_781,239_209_529,MInt2113929217,],[Modulo1811939329,1_811_939_329,1_811_939_327,671_088_638,959_408_210,1_483_943_592,MInt1811939329,],[Modulo2013265921,2_013_265_921,2_013_265_919,268_435_454,1_172_168_163,317_946_875,MInt2013265921,],);}
pub use self::zero_one::{One,Zero};
mod zero_one{pub trait Zero:Sized{fn zero()->Self;#[inline]fn is_zero(&self)->bool where Self:PartialEq{self==&Self::zero()}#[inline]fn set_zero(&mut self){*self=Self::zero();}}pub trait One:Sized{fn one()->Self;#[inline]fn is_one(&self)->bool where Self:PartialEq{self==&Self::one()}#[inline]fn set_one(&mut self){*self=Self::one();}}macro_rules!zero_one_impls{($({$Trait:ident$method:ident$($t:ty)*,$e:expr})*)=>{$($(impl$Trait for$t{fn$method()->Self{$e}})*)*};}zero_one_impls!({Zero zero u8 u16 u32 u64 usize i8 i16 i32 i64 isize u128 i128,0}{Zero zero f32 f64,0.}{One one u8 u16 u32 u64 usize i8 i16 i32 i64 isize u128 i128,1}{One one f32 f64,1.});}
pub use self::rolling_hash::{Gf2_63x1,Gf2_63x2,Gf2_63x3,HashedRangeChained,Mersenne61x1,Mersenne61x2,Mersenne61x3,RollingHasher};
pub mod rolling_hash{use super::{Gf2_63,Invertible,Mersenne61,Ring,SemiRing,Xorshift};use std::{cmp::Ordering,fmt::{self,Debug},marker::PhantomData,ops::{Bound,RangeBounds,RangeInclusive}};pub trait RollingHasher{type T;type Hash:Copy+Eq;fn init(len:usize,rng:&mut Xorshift);fn init_with_time(len:usize){let mut rng=Xorshift::time();Self::init(len,&mut rng);}fn ensure(len:usize);fn hash_sequence<I>(iter:I)->HashedSequence<Self>where I:IntoIterator<Item=Self::T>;fn hash_substr(hashed:&[Self::Hash])->Hashed<Self>;fn concat_hash(x:&Hashed<Self>,y:&Hashed<Self>)->Hashed<Self>;}#[derive(Debug)]pub struct HashedSequence<Hasher>where Hasher:RollingHasher+?Sized{hashed:Vec<Hasher::Hash>,_marker:PhantomData<fn()->Hasher>}impl<Hasher>HashedSequence<Hasher>where Hasher:RollingHasher+?Sized{fn new(hashed:Vec<Hasher::Hash>)->Self{Self{hashed,_marker:PhantomData}}pub fn len(&self)->usize{self.hashed.len()-1}pub fn is_empty(&self)->bool{self.len()==0}pub fn range<R>(&self,range:R)->HashedRange<'_,Hasher>where R:RangeBounds<usize>{HashedRange::new(&self.hashed[to_range(range,self.len())])}pub fn hash_range<R>(&self,range:R)->Hashed<Hasher>where R:RangeBounds<usize>{self.range(range).hash()}}#[derive(Debug)]pub struct HashedRange<'a,Hasher>where Hasher:RollingHasher+?Sized{hashed:&'a[Hasher::Hash],_marker:PhantomData<fn()->Hasher>}impl<Hasher>Clone for HashedRange<'_,Hasher>where Hasher:RollingHasher+?Sized{fn clone(&self)->Self{Self{hashed:self.hashed,_marker:self._marker}}}impl<Hasher>Copy for HashedRange<'_,Hasher>where Hasher:RollingHasher+?Sized{}impl<Hasher>PartialEq for HashedRange<'_,Hasher>where Hasher:RollingHasher+?Sized{fn eq(&self,other:&Self)->bool{self.hash()==other.hash()}}impl<Hasher>Eq for HashedRange<'_,Hasher>where Hasher:RollingHasher+?Sized{}impl<Hasher>PartialOrd for HashedRange<'_,Hasher>where Hasher:RollingHasher+?Sized,Hasher::Hash:PartialOrd{fn partial_cmp(&self,other:&Self)->Option<Ordering>{let n=self.longest_common_prefix(other);match(self.len()>n,other.len()>n){(true,true)=>{let x=self.hash_range(n..=n);let y=other.hash_range(n..=n);x.hash.partial_cmp(&y.hash)}(x,y)=>Some(x.cmp(&y)),}}}impl<Hasher>Ord for HashedRange<'_,Hasher>where Hasher:RollingHasher+?Sized,Hasher::Hash:Ord{fn cmp(&self,other:&Self)->Ordering{let n=self.longest_common_prefix(other);match(self.len()>n,other.len()>n){(true,true)=>{let x=self.hash_range(n..=n);let y=other.hash_range(n..=n);x.hash.cmp(&y.hash)}(x,y)=>x.cmp(&y),}}}impl<'a,Hasher>HashedRange<'a,Hasher>where Hasher:RollingHasher+?Sized{fn new(hashed:&'a[Hasher::Hash])->Self{Self{hashed,_marker:PhantomData}}pub fn len(&self)->usize{self.hashed.len()-1}pub fn is_empty(&self)->bool{self.len()==0}pub fn range<R>(&self,range:R)->HashedRange<'a,Hasher>where R:RangeBounds<usize>{HashedRange::new(&self.hashed[to_range(range,self.len())])}pub fn hash_range<R>(&self,range:R)->Hashed<Hasher>where R:RangeBounds<usize>{self.range(range).hash()}pub fn hash(&self)->Hashed<Hasher>{Hasher::hash_substr(self.hashed)}pub fn longest_common_prefix(&self,other:&Self)->usize{let n=self.len().min(other.len());let mut ok=0usize;let mut err=n+1;while ok+1<err{let mid=(ok+err)/2;if self.range(..mid).hash()==other.range(..mid).hash(){ok=mid;}else{err=mid;}}ok}pub fn chainable(self)->HashedRangeChained<'a,Hasher>{vec![self].into()}}pub struct HashedRangeChained<'a,Hasher>where Hasher:RollingHasher+?Sized{chained:Vec<HashedRange<'a,Hasher>>,_marker:PhantomData<fn()->Hasher>}impl<'a,Hasher:Debug>Debug for HashedRangeChained<'a,Hasher>where Hasher:RollingHasher+?Sized,Hasher::Hash:Debug{fn fmt(&self,f:&mut fmt::Formatter<'_>)->fmt::Result{f.debug_struct("HashedRangeChained").field("chained",&self.chained).finish()}}impl<'a,Hasher:Default>Default for HashedRangeChained<'a,Hasher>where Hasher:RollingHasher+?Sized{fn default()->Self{Self{chained:Default::default(),_marker:Default::default()}}}impl<'a,Hasher:Clone>Clone for HashedRangeChained<'a,Hasher>where Hasher:RollingHasher+?Sized{fn clone(&self)->Self{Self{chained:self.chained.clone(),_marker:self._marker}}}impl<Hasher>PartialEq for HashedRangeChained<'_,Hasher>where Hasher:RollingHasher+?Sized{fn eq(&self,other:&Self)->bool{let mut a=self.chained.iter().cloned();let mut b=other.chained.iter().cloned();macro_rules!next{($iter:expr)=>{loop{if let Some(x)=$iter.next(){if x.len()>0{break Some(x);}}else{break None;}}};}let mut x:Option<HashedRange<'_,Hasher>>=None;let mut y:Option<HashedRange<'_,Hasher>>=None;loop{if x.map_or(true,|x|x.is_empty()){x=next!(a);}if y.map_or(true,|y|y.is_empty()){y=next!(b);}if let(Some(x),Some(y))=(&mut x,&mut y){let k=x.len().min(y.len());if x.range(..k)!=y.range(..k){return false;}*x=x.range(k..);*y=y.range(k..);}else{break x.is_none()==y.is_none();}}}}impl<Hasher>Eq for HashedRangeChained<'_,Hasher>where Hasher:RollingHasher+?Sized{}impl<Hasher>PartialOrd for HashedRangeChained<'_,Hasher>where Hasher:RollingHasher+?Sized,Hasher::Hash:PartialOrd{fn partial_cmp(&self,other:&Self)->Option<Ordering>{let mut a=self.chained.iter().cloned();let mut b=other.chained.iter().cloned();macro_rules!next{($iter:expr)=>{loop{if let Some(x)=$iter.next(){if x.len()>0{break Some(x);}}else{break None;}}};}let mut x:Option<HashedRange<'_,Hasher>>=None;let mut y:Option<HashedRange<'_,Hasher>>=None;loop{if x.map_or(true,|x|x.is_empty()){x=next!(a);}if y.map_or(true,|y|y.is_empty()){y=next!(b);}if let(Some(x),Some(y))=(&mut x,&mut y){let k=x.longest_common_prefix(y);if x.len()>k&&y.len()>k{let x=x.hash_range(k..=k);let y=y.hash_range(k..=k);break x.hash.partial_cmp(&y.hash);};*x=x.range(k..);*y=y.range(k..);}else{break x.is_some().partial_cmp(&y.is_some());}}}}impl<Hasher>Ord for HashedRangeChained<'_,Hasher>where Hasher:RollingHasher+?Sized,Hasher::Hash:Ord{fn cmp(&self,other:&Self)->Ordering{let mut a=self.chained.iter().cloned();let mut b=other.chained.iter().cloned();macro_rules!next{($iter:expr)=>{loop{if let Some(x)=$iter.next(){if x.len()>0{break Some(x);}}else{break None;}}};}let mut x:Option<HashedRange<'_,Hasher>>=None;let mut y:Option<HashedRange<'_,Hasher>>=None;loop{if x.map_or(true,|x|x.is_empty()){x=next!(a);}if y.map_or(true,|y|y.is_empty()){y=next!(b);}if let(Some(x),Some(y))=(&mut x,&mut y){let k=x.longest_common_prefix(y);if x.len()>k&&y.len()>k{let x=x.hash_range(k..=k);let y=y.hash_range(k..=k);break x.hash.cmp(&y.hash);};*x=x.range(k..);*y=y.range(k..);}else{break x.is_some().cmp(&y.is_some());}}}}impl<'a,Hasher>From<Vec<HashedRange<'a,Hasher>>>for HashedRangeChained<'a,Hasher>where Hasher:RollingHasher+?Sized{fn from(hashed:Vec<HashedRange<'a,Hasher>>)->Self{Self{chained:hashed,_marker:PhantomData}}}impl<'a,Hasher>HashedRangeChained<'a,Hasher>where Hasher:RollingHasher+?Sized{pub fn chain(mut self,x:HashedRange<'a,Hasher>)->Self{self.chained.push(x);self}pub fn push(&mut self,x:HashedRange<'a,Hasher>){self.chained.push(x);}}fn to_range<R>(range:R,ub:usize)->RangeInclusive<usize>where R:RangeBounds<usize>{let l=match range.start_bound(){Bound::Included(l)=>*l,Bound::Excluded(l)=>l+1,Bound::Unbounded=>0,};let r=match range.end_bound(){Bound::Included(r)=>r+1,Bound::Excluded(r)=>*r,Bound::Unbounded=>ub,};l..=r}#[derive(Debug)]pub struct Hashed<Hasher>where Hasher:RollingHasher+?Sized{len:usize,hash:Hasher::Hash,_marker:PhantomData<fn()->Hasher>}impl<Hasher>std::hash::Hash for Hashed<Hasher>where Hasher:RollingHasher+?Sized,Hasher::Hash:std::hash::Hash{fn hash<H:std::hash::Hasher>(&self,state:&mut H){self.len.hash(state);self.hash.hash(state);self._marker.hash(state);}}impl<Hasher>Hashed<Hasher>where Hasher:RollingHasher+?Sized{fn new(len:usize,hash:Hasher::Hash)->Self{Self{len,hash,_marker:PhantomData}}pub fn concat(&self,other:&Self)->Self{Hasher::concat_hash(self,other)}}impl<Hasher>Clone for Hashed<Hasher>where Hasher:RollingHasher+?Sized{fn clone(&self)->Self{Self{len:self.len,hash:self.hash,_marker:self._marker}}}impl<Hasher>Copy for Hashed<Hasher>where Hasher:RollingHasher+?Sized{}impl<Hasher>PartialEq for Hashed<Hasher>where Hasher:RollingHasher+?Sized{fn eq(&self,other:&Self)->bool{self.len==other.len&&self.hash==other.hash}}impl<Hasher>Eq for Hashed<Hasher>where Hasher:RollingHasher+?Sized{}#[derive(Debug)]struct RollingHashPrecalc<R>where R:SemiRing{base:R::T,pow:Vec<R::T>}impl<R>Default for RollingHashPrecalc<R>where R:SemiRing,R::T:Default{fn default()->Self{Self{base:Default::default(),pow:Default::default()}}}impl<R>RollingHashPrecalc<R>where R:SemiRing,R::Additive:Invertible{fn new(base:R::T)->Self{Self{base,pow:vec![R::one()]}}fn ensure_pow(&mut self,len:usize){if self.pow.len()<=len{self.pow.reserve(len-self.pow.len()+1);if self.pow.is_empty(){self.pow.push(R::one());}for _ in 0..=len-self.pow.len(){self.pow.push(R::mul(self.pow.last().unwrap(),&self.base));}}}fn mul1_add(&self,x:&R::T,y:&R::T)->R::T{R::add(&R::mul(x,&self.base),y)}fn muln_add(&mut self,x:&R::T,y:&R::T,n:usize)->R::T{R::add(&R::mul(x,&self.pow[n]),y)}fn muln_sub(&mut self,l:&R::T,r:&R::T,n:usize)->R::T{R::sub(r,&R::mul(l,&self.pow[n]))}}macro_rules!impl_rolling_hasher{(@inner$T:ident,$R:ty,[$($i:tt)*][$($s:tt)*][$a:tt$($tt:tt)*][$k:tt$($j:tt)*])=>{impl_rolling_hasher!(@inner$T,$R,[$($i)*$k][$($s)*()][$($tt)*][$($j)*]);};(@inner$T:ident,$R:ty,[$($i:tt)+][$($s:tt)+][][$len:tt$($j:tt)*])=>{#[derive(Debug,Clone,Copy,Hash,PartialEq,Eq,PartialOrd,Ord)]pub enum$T{}impl$T{fn __rolling_hash_local_key()->&'static::std::thread::LocalKey<::std::cell::Cell<[RollingHashPrecalc<$R>;$len]>>{::std::thread_local!(static __LOCAL_KEY: ::std::cell::Cell<[RollingHashPrecalc<$R>;$len]> =::std::cell::Cell::new(Default::default()));&__LOCAL_KEY}}impl RollingHasher for$T{type T=<$R as SemiRing>::T;type Hash=[<$R as SemiRing>::T;$len];fn init(len:usize,rng:&mut Xorshift){Self::__rolling_hash_local_key().with(|cell|{if unsafe{(&*cell.as_ptr()).iter().all(|p|p.base==0)}{cell.set([$({$s;RollingHashPrecalc::new(rng.rand(<$R>::MOD))},)+]);}});Self::ensure(len);}fn ensure(len:usize){Self::__rolling_hash_local_key().with(|cell|{unsafe{let arr=&mut*cell.as_ptr();$(arr[$i].ensure_pow(len);)+}})}fn hash_sequence<I>(iter:I)->HashedSequence<Self>where I:IntoIterator<Item=Self::T>,{let iter=iter.into_iter();let(lb,_)=iter.size_hint();let mut hashed=Vec::with_capacity(lb+1);hashed.push([$({$s;<$R>::zero()},)+]);unsafe{Self::__rolling_hash_local_key().with(|cell|{let arr=&*cell.as_ptr();for item in iter{let last=hashed.last().unwrap();let h=[$(arr[$i].mul1_add(&last[$i],&item),)+];hashed.push(h);}})};HashedSequence::new(hashed)}fn hash_substr(hashed:&[Self::Hash])->Hashed<Self>{let len=hashed.len()-1;let h=unsafe{Self::__rolling_hash_local_key().with(|cell|{let arr=&mut*cell.as_ptr();[$(arr[$i].muln_sub(&hashed[0][$i],&hashed[len][$i],len),)+]})};Hashed::new(len,h)}fn concat_hash(x:&Hashed<Self>,y:&Hashed<Self>)->Hashed<Self>{let len=y.len;let hash=unsafe{Self::__rolling_hash_local_key().with(|cell|{let arr=&mut*cell.as_ptr();[$(arr[$i].muln_add(&x.hash[$i],&y.hash[$i],len),)+]})};Hashed::new(x.len+y.len,hash)}}};($T:ident,$R:ty,[$($tt:tt)+])=>{impl_rolling_hasher!(@inner$T,$R,[][][$($tt)+][0 1 2 3 4 5 6 7 8 9]);};}impl_rolling_hasher!(Mersenne61x1,Mersenne61,[_]);impl_rolling_hasher!(Mersenne61x2,Mersenne61,[_ _]);impl_rolling_hasher!(Mersenne61x3,Mersenne61,[_ _ _]);impl_rolling_hasher!(Gf2_63x1,Gf2_63,[_]);impl_rolling_hasher!(Gf2_63x2,Gf2_63,[_ _]);impl_rolling_hasher!(Gf2_63x3,Gf2_63,[_ _ _]);}
pub use self::additive_operation_impl::AdditiveOperation;
mod additive_operation_impl{use super::*;use std::{marker::PhantomData,ops::{Add,Neg,Sub}};#[doc=" $+$"]pub struct AdditiveOperation<T:Clone+Zero+Add<Output=T>>{_marker:PhantomData<fn()->T>}impl<T:Clone+Zero+Add<Output=T>>Magma for AdditiveOperation<T>{type T=T;#[inline]fn operate(x:&Self::T,y:&Self::T)->Self::T{x.clone()+y.clone()}}impl<T:Clone+Zero+Add<Output=T>>Unital for AdditiveOperation<T>{#[inline]fn unit()->Self::T{Zero::zero()}}impl<T:Clone+Zero+Add<Output=T>>Associative for AdditiveOperation<T>{}impl<T:Clone+Zero+Add<Output=T>>Commutative for AdditiveOperation<T>{}impl<T:Clone+Zero+Add<Output=T>+Sub<Output=T>+Neg<Output=T>>Invertible for AdditiveOperation<T>{#[inline]fn inverse(x:&Self::T)->Self::T{-x.clone()}#[inline]fn rinv_operate(x:&Self::T,y:&Self::T)->Self::T{x.clone()-y.clone()}}}
pub use self::bitxor_operation_impl::{BitXorIdentity,BitXorOperation};
mod bitxor_operation_impl{use super::*;use std::{marker::PhantomData,ops::BitXor};#[doc=" ^"]pub struct BitXorOperation<T:Clone+BitXorIdentity>{_marker:PhantomData<fn()->T>}pub trait BitXorIdentity:Sized+BitXor<Output=Self>{fn xor_zero()->Self;}#[macro_export]macro_rules!impl_bitxor_identity{([$($wh:tt)*],$t:ty,$xor_zero:expr)=>{impl<$($wh)*>BitXorIdentity for$t{#[inline]fn xor_zero()->Self{$xor_zero}}};($t:ty,$xor_zero:expr)=>{impl BitXorIdentity for$t{#[inline]fn xor_zero()->Self{$xor_zero}}};}impl_bitxor_identity!(bool,false);impl_bitxor_identity!(usize,0usize);impl_bitxor_identity!(u8,0u8);impl_bitxor_identity!(u16,0u16);impl_bitxor_identity!(u32,0u32);impl_bitxor_identity!(u64,0u64);impl_bitxor_identity!(isize,0isize);impl_bitxor_identity!(i8,0i8);impl_bitxor_identity!(i16,0i16);impl_bitxor_identity!(i32,0i32);impl_bitxor_identity!(i64,0i64);impl<T:Clone+BitXorIdentity>Magma for BitXorOperation<T>{type T=T;#[inline]fn operate(x:&Self::T,y:&Self::T)->Self::T{x.clone()^y.clone()}}impl<T:Clone+BitXorIdentity>Unital for BitXorOperation<T>{#[inline]fn unit()->Self::T{BitXorIdentity::xor_zero()}}impl<T:Clone+BitXorIdentity>Associative for BitXorOperation<T>{}impl<T:Clone+BitXorIdentity>Commutative for BitXorOperation<T>{}impl<T:Clone+BitXorIdentity>Invertible for BitXorOperation<T>{fn inverse(x:&Self::T)->Self::T{x.clone()}}}
pub use self::gf2_63::Gf2_63;
mod gf2_63{use super::*;pub enum Gf2_63{}impl Gf2_63{pub const MOD:u64=1<<63;}impl Magma for Gf2_63{type T=u64;fn operate(x:&Self::T,y:&Self::T)->Self::T{use core::arch::x86_64::{_mm_clmulepi64_si128,_mm_extract_epi64,_mm_set_epi64x};unsafe{let a=_mm_set_epi64x(0,*x as i64);let b=_mm_set_epi64x(0,*y as i64);let c=_mm_clmulepi64_si128(a,b,0);let lo=_mm_extract_epi64(c,0)as u64;let hi=_mm_extract_epi64(c,1)as u64;let hi=hi<<1|lo>>63;let lo=lo&!(!(0u64)<<63);lo^hi^(hi<<1)}}}impl Unital for Gf2_63{fn unit()->Self::T{1}}impl Associative for Gf2_63{}impl Commutative for Gf2_63{}impl SemiRing for Gf2_63{type T=u64;type Additive=BitXorOperation<u64>;type Multiplicative=Self;}}
pub use self::mersenne61::Mersenne61;
mod mersenne61{use super::*;pub enum Mersenne61Add{}impl Magma for Mersenne61Add{type T=u64;fn operate(x:&Self::T,y:&Self::T)->Self::T{let mut z=x+y;if z>=Mersenne61::MOD{z-=Mersenne61::MOD}z}}impl Unital for Mersenne61Add{fn unit()->Self::T{0}}impl Associative for Mersenne61Add{}impl Commutative for Mersenne61Add{}impl Invertible for Mersenne61Add{fn inverse(x:&Self::T)->Self::T{if*x==0{0}else{Mersenne61::MOD-x}}}pub enum Mersenne61{}impl Mersenne61{pub const MOD:u64=(1<<61)-1;}impl Magma for Mersenne61{type T=u64;fn operate(x:&Self::T,y:&Self::T)->Self::T{let z=*x as u128**y as u128;Mersenne61Add::operate(&((z>>61)as _),&(z as u64&Self::MOD))}}impl Unital for Mersenne61{fn unit()->Self::T{1}}impl Associative for Mersenne61{}impl Commutative for Mersenne61{}impl SemiRing for Mersenne61{type T=u64;type Additive=Mersenne61Add;type Multiplicative=Self;}}
pub use self::multiplicative_operation_impl::MultiplicativeOperation;
mod multiplicative_operation_impl{use super::*;use std::{marker::PhantomData,ops::{Div,Mul}};#[doc=" $\\times$"]pub struct MultiplicativeOperation<T:Clone+One+Mul<Output=T>>{_marker:PhantomData<fn()->T>}impl<T:Clone+One+Mul<Output=T>>Magma for MultiplicativeOperation<T>{type T=T;#[inline]fn operate(x:&Self::T,y:&Self::T)->Self::T{x.clone()*y.clone()}}impl<T:Clone+One+Mul<Output=T>>Unital for MultiplicativeOperation<T>{#[inline]fn unit()->Self::T{One::one()}}impl<T:Clone+One+Mul<Output=T>>Associative for MultiplicativeOperation<T>{}impl<T:Clone+One+Mul<Output=T>>Commutative for MultiplicativeOperation<T>{}impl<T:Clone+One+Mul<Output=T>+Div<Output=T>>Invertible for MultiplicativeOperation<T>{#[inline]fn inverse(x:&Self::T)->Self::T{Self::unit().div(x.clone())}#[inline]fn rinv_operate(x:&Self::T,y:&Self::T)->Self::T{(x.clone()).div(y.clone())}}}
#[derive(Clone,Debug)]pub struct Xorshift{y:u64}
impl Xorshift{pub fn new(seed:u64)->Self{Xorshift{y:seed}}pub fn time()->Self{let seed=std::time::SystemTime::now().duration_since(std::time::UNIX_EPOCH).ok().unwrap_or_default().as_secs();Xorshift::new(seed)}#[inline]pub fn rand64(&mut self)->u64{self.y^=self.y<<5;self.y^=self.y>>17;self.y^=self.y<<11;self.y}#[inline]pub fn rand(&mut self,k:u64)->u64{self.rand64()%k}#[inline]pub fn rands(&mut self,k:u64,n:usize)->Vec<u64>{(0..n).map(|_|self.rand(k)).collect::<Vec<_>>()}#[inline]pub fn randf(&mut self)->f64{const UPPER_MASK:u64=0x3FF0_0000_0000_0000;const LOWER_MASK:u64=0x000F_FFFF_FFFF_FFFF;let tmp=UPPER_MASK|(self.rand64()&LOWER_MASK);let result:f64=f64::from_bits(tmp);result-1.0}#[inline]pub fn gen_bool(&mut self,p:f64)->bool{self.randf()<p}pub fn shuffle<T>(&mut self,slice:&mut[T]){let mut n=slice.len();while n>1{let i=self.rand(n as _)as usize;n-=1;slice.swap(i,n);}}}
impl Default for Xorshift{fn default()->Self{Xorshift::new(0x2b99_2ddf_a232_49d6)}}
pub use self::magma::*;
mod magma{#![doc=" algebraic traits"]#[doc=" binary operaion: $T \\circ T \\to T$"]pub trait Magma{#[doc=" type of operands: $T$"]type T:Clone;#[doc=" binary operaion: $\\circ$"]fn operate(x:&Self::T,y:&Self::T)->Self::T;#[inline]fn reverse_operate(x:&Self::T,y:&Self::T)->Self::T{Self::operate(y,x)}#[inline]fn operate_assign(x:&mut Self::T,y:&Self::T){*x=Self::operate(x,y);}}#[doc=" $\\forall a,\\forall b,\\forall c \\in T, (a \\circ b) \\circ c = a \\circ (b \\circ c)$"]pub trait Associative{}#[doc=" associative binary operation"]pub trait SemiGroup:Magma+Associative{}impl<S>SemiGroup for S where S:Magma+Associative{}#[doc=" $\\exists e \\in T, \\forall a \\in T, e \\circ a = a \\circ e = e$"]pub trait Unital:Magma{#[doc=" identity element: $e$"]fn unit()->Self::T;#[inline]fn is_unit(x:&Self::T)->bool where<Self as Magma>::T:PartialEq{x==&Self::unit()}#[inline]fn set_unit(x:&mut Self::T){*x=Self::unit();}}#[doc=" associative binary operation and an identity element"]pub trait Monoid:SemiGroup+Unital{#[doc=" binary exponentiation: $x^n = x\\circ\\ddots\\circ x$"]fn pow(mut x:Self::T,mut n:usize)->Self::T{let mut res=Self::unit();while n>0{if n&1==1{res=Self::operate(&res,&x);}x=Self::operate(&x,&x);n>>=1;}res}}impl<M>Monoid for M where M:SemiGroup+Unital{}#[doc=" $\\exists e \\in T, \\forall a \\in T, \\exists b,c \\in T, b \\circ a = a \\circ c = e$"]pub trait Invertible:Magma{#[doc=" $a$ where $a \\circ x = e$"]fn inverse(x:&Self::T)->Self::T;#[inline]fn rinv_operate(x:&Self::T,y:&Self::T)->Self::T{Self::operate(x,&Self::inverse(y))}}#[doc=" associative binary operation and an identity element and inverse elements"]pub trait Group:Monoid+Invertible{}impl<G>Group for G where G:Monoid+Invertible{}#[doc=" $\\forall a,\\forall b \\in T, a \\circ b = b \\circ a$"]pub trait Commutative{}#[doc=" commutative monoid"]pub trait AbelianMonoid:Monoid+Commutative{}impl<M>AbelianMonoid for M where M:Monoid+Commutative{}#[doc=" commutative group"]pub trait AbelianGroup:Group+Commutative{}impl<G>AbelianGroup for G where G:Group+Commutative{}#[doc=" $\\forall a \\in T, a \\circ a = a$"]pub trait Idempotent{}#[doc=" idempotent monoid"]pub trait IdempotentMonoid:Monoid+Idempotent{}impl<M>IdempotentMonoid for M where M:Monoid+Idempotent{}#[macro_export]macro_rules!monoid_fold{($m:ty)=>{<$m as Unital>::unit()};($m:ty,)=>{<$m as Unital>::unit()};($m:ty,$f:expr)=>{$f};($m:ty,$f:expr,$($ff:expr),*)=>{<$m as Magma>::operate(&($f),&monoid_fold!($m,$($ff),*))};}#[macro_export]macro_rules!define_monoid{($Name:ident,$t:ty,|$x:ident,$y:ident|$op:expr,$unit:expr)=>{struct$Name;impl Magma for$Name{type T=$t;fn operate($x:&Self::T,$y:&Self::T)->Self::T{$op}}impl Unital for$Name{fn unit()->Self::T{$unit}}impl Associative for$Name{}};}}
pub use self::ring::*;
mod ring{use super::*;use std::{marker::PhantomData,ops::{Add,Mul}};pub trait SemiRing{type T:Clone;type Additive:AbelianMonoid<T=Self::T>;type Multiplicative:Monoid<T=Self::T>;#[doc=" additive identity: $0$"]fn zero()->Self::T{<Self::Additive as Unital>::unit()}#[doc=" multiplicative identity: $1$"]fn one()->Self::T{<Self::Multiplicative as Unital>::unit()}#[doc=" additive operaion: $+$"]fn add(x:&Self::T,y:&Self::T)->Self::T{<Self::Additive as Magma>::operate(x,y)}#[doc=" multiplicative operaion: $+$"]fn mul(x:&Self::T,y:&Self::T)->Self::T{<Self::Multiplicative as Magma>::operate(x,y)}}pub trait Ring:SemiRing where Self::Additive:Invertible{#[doc=" additive inverse: $-$"]fn neg(x:&Self::T)->Self::T{<Self::Additive as Invertible>::inverse(x)}#[doc=" additive right inversed operaion: $-$"]fn sub(x:&Self::T,y:&Self::T)->Self::T{Self::add(x,&Self::neg(y))}}impl<R>Ring for R where R:SemiRing,R::Additive:Invertible{}pub trait Field:Ring where Self::Additive:Invertible,Self::Multiplicative:Invertible{#[doc=" multiplicative inverse: $-$"]fn inv(x:&Self::T)->Self::T{<Self::Additive as Invertible>::inverse(x)}#[doc=" multiplicative right inversed operaion: $-$"]fn div(x:&Self::T,y:&Self::T)->Self::T{Self::mul(x,&Self::inv(y))}}impl<F>Field for F where F:Ring,F::Additive:Invertible,F::Multiplicative:Invertible{}#[doc=" $+,\\times$"]pub struct AddMulOperation<T>where T:Clone+Zero+One+Add<Output=T>+Mul<Output=T>{_marker:PhantomData<fn()->T>}impl<T>SemiRing for AddMulOperation<T>where T:Clone+Zero+One+Add<Output=T>+Mul<Output=T>{type T=T;type Additive=AdditiveOperation<T>;type Multiplicative=MultiplicativeOperation<T>;}}