pub fn main() {
    crate::prepare!();
    sc!(n, m, l: [usize; m]);
    let mut x = vec![M::zero(); n + 1];
    x[0] = M::one();
    for &l in &l {
        x[l] += M::one();
    }
    let ans = Fps998244353::from_vec(x).pow(n + 1, n + 1)[n] / M::from(n + 1);
    pp!(ans);
}
pub type M = mint_basic::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)*);}};}}
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)*$U,$($Rest)*);};($($T:ident)*)=>{iter_scan_tuple_impl!(@inner,$($T)*);};}iter_scan_tuple_impl!(A B C D E F G H I J K);pub struct ScannerIter<'a,'b,T>{inner:&'b mut Scanner<'a>,_marker:std::marker::PhantomData<fn()->T>}impl<'a,'b,T>Iterator for ScannerIter<'a,'b,T>where T:IterScan{type Item=<T as IterScan>::Output;#[inline]fn next(&mut self)->Option<Self::Item>{<T as IterScan>::scan(&mut self.inner.iter)}}#[doc=" - `scan_value!(scanner, ELEMENT)`"]#[doc=""]#[doc=" ELEMENT :="]#[doc=" - `$ty`: IterScan"]#[doc=" - `@$expr`: MarkedIterScan"]#[doc=" - `[ELEMENT; $expr]`: vector"]#[doc=" - `[ELEMENT]`: iterator"]#[doc=" - `($(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($len).collect::<Vec<_>>())?};(@tuple$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;$len:expr]$($t:tt)*)=>{$crate::scan_value!(@$tag$scanner,[$($args)*[$crate::scan_value!(@repeat$scanner,[@$e]$len)]]$($t)*)};(@$tag:ident$scanner:expr,[$($args:tt)*][[$($tt:tt)*];$len:expr]$($t:tt)*)=>{$crate::scan_value!(@$tag$scanner,[$($args)*[$crate::scan_value!(@repeat$scanner,[[$($tt)*]]$len)]]$($t)*)};(@$tag:ident$scanner:expr,[$($args:tt)*][($($tt:tt)*);$len:expr]$($t:tt)*)=>{$crate::scan_value!(@$tag$scanner,[$($args)*[$crate::scan_value!(@repeat$scanner,[($($tt)*)]$len)]]$($t)*)};(@$tag:ident$scanner:expr,[$($args:tt)*][$ty:ty;$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!(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 struct 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 struct 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 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 struct 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()?)}}}
#[derive(Debug,Default)]pub struct FormalPowerSeries<T,C>{pub data:Vec<T>,_marker:std::marker::PhantomData<C>}
pub type Fps998244353=FormalPowerSeries<mint_basic::MInt998244353,Convolve998244353>;
pub type Fps<M>=FormalPowerSeries<MInt<M>,MIntConvolve<M>>;
pub trait FormalPowerSeriesCoefficient:Sized+Clone+Zero+PartialEq+One+From<usize>+std::ops::Add<Output=Self>+std::ops::Sub<Output=Self>+std::ops::Mul<Output=Self>+std::ops::Div<Output=Self>+for<'r>std::ops::Add<&'r Self,Output=Self>+for<'r>std::ops::Sub<&'r Self,Output=Self>+for<'r>std::ops::Mul<&'r Self,Output=Self>+for<'r>std::ops::Div<&'r Self,Output=Self>+std::ops::AddAssign<Self>+std::ops::SubAssign<Self>+std::ops::MulAssign<Self>+std::ops::DivAssign<Self>+for<'r>std::ops::AddAssign<&'r Self>+for<'r>std::ops::SubAssign<&'r Self>+for<'r>std::ops::MulAssign<&'r Self>+for<'r>std::ops::DivAssign<&'r Self>+std::ops::Neg<Output=Self>{}
impl<M>FormalPowerSeriesCoefficient for MInt<M>where M:MIntConvert<usize>{}
pub trait FormalPowerSeriesCoefficientSqrt:FormalPowerSeriesCoefficient{fn sqrt_coefficient(&self)->Option<Self>;}
impl<M:MIntConvert<u32>+MIntConvert<usize>>FormalPowerSeriesCoefficientSqrt for MInt<M>{fn sqrt_coefficient(&self)->Option<Self>{self.sqrt()}}
mod formal_power_series_impls{use super::*;use std::{iter::repeat_with,iter::{once,FromIterator},marker::PhantomData,ops::{Index,IndexMut},slice::{Iter,IterMut}};impl<T,C>FormalPowerSeries<T,C>{pub fn from_vec(data:Vec<T>)->Self{Self{data,_marker:PhantomData}}pub fn length(&self)->usize{self.data.len()}pub fn truncate(&mut self,deg:usize){self.data.truncate(deg)}pub fn iter(&self)->Iter<'_,T>{self.data.iter()}pub fn iter_mut(&mut self)->IterMut<'_,T>{self.data.iter_mut()}}impl<T,C>Clone for FormalPowerSeries<T,C>where T:Clone{fn clone(&self)->Self{Self::from_vec(self.data.clone())}}impl<T,C>PartialEq for FormalPowerSeries<T,C>where T:PartialEq{fn eq(&self,other:&Self)->bool{self.data.eq(&other.data)}}impl<T,C>Eq for FormalPowerSeries<T,C>where T:PartialEq{}impl<T,C>FormalPowerSeries<T,C>where T:Zero{pub fn zeros(deg:usize)->Self{repeat_with(T::zero).take(deg).collect()}pub fn resize(&mut self,deg:usize){self.data.resize_with(deg,Zero::zero)}pub fn resized(mut self,deg:usize)->Self{self.resize(deg);self}pub fn reversed(mut self)->Self{self.data.reverse();self}}impl<T,C>FormalPowerSeries<T,C>where T:Zero+PartialEq{pub fn trim_tail_zeros(&mut self){let mut len=self.length();while len>0{if self.data[len-1].is_zero(){len-=1;}else{break;}}self.truncate(len);}}impl<T,C>Zero for FormalPowerSeries<T,C>where T:PartialEq{fn zero()->Self{Self::from_vec(Vec::new())}}impl<T,C>One for FormalPowerSeries<T,C>where T:PartialEq+One{fn one()->Self{Self::from(T::one())}}impl<T,C>IntoIterator for FormalPowerSeries<T,C>{type Item=T;type IntoIter=std::vec::IntoIter<T>;fn into_iter(self)->Self::IntoIter{self.data.into_iter()}}impl<T,C>FromIterator<T>for FormalPowerSeries<T,C>{fn from_iter<I:IntoIterator<Item=T>>(iter:I)->Self{Self::from_vec(iter.into_iter().collect())}}impl<T,C>Index<usize>for FormalPowerSeries<T,C>{type Output=T;fn index(&self,index:usize)->&Self::Output{&self.data[index]}}impl<T,C>IndexMut<usize>for FormalPowerSeries<T,C>{fn index_mut(&mut self,index:usize)->&mut Self::Output{&mut self.data[index]}}impl<T,C>From<T>for FormalPowerSeries<T,C>{fn from(x:T)->Self{once(x).collect()}}impl<T,C>From<Vec<T>>for FormalPowerSeries<T,C>{fn from(data:Vec<T>)->Self{Self::from_vec(data)}}impl<T,C>FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient{pub fn prefix_ref(&self,deg:usize)->Self{if deg<self.length(){Self::from_vec(self.data[..deg].to_vec())}else{self.clone()}}pub fn prefix(mut self,deg:usize)->Self{self.data.truncate(deg);self}pub fn even(mut self)->Self{let mut keep=false;self.data.retain(|_|{keep=!keep;keep});self}pub fn odd(mut self)->Self{let mut keep=true;self.data.retain(|_|{keep=!keep;keep});self}pub fn diff(mut self)->Self{let mut c=T::one();for x in self.iter_mut().skip(1){*x*=&c;c+=T::one();}if self.length()>0{self.data.remove(0);}self}pub fn integral(mut self)->Self{let n=self.length();self.data.insert(0,Zero::zero());let mut fact=Vec::with_capacity(n+1);let mut c=T::one();fact.push(c.clone());for _ in 1..n{fact.push(fact.last().cloned().unwrap()*c.clone());c+=T::one();}let mut invf=T::one()/(fact.last().cloned().unwrap()*c.clone());for x in self.iter_mut().skip(1).rev(){*x*=invf.clone()*fact.pop().unwrap();invf*=c.clone();c-=T::one();}self}pub fn eval(&self,x:T)->T{let mut base=T::one();let mut res=T::zero();for a in self.iter(){res+=base.clone()*a.clone();base*=x.clone();}res}}impl<T,C>FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient,C:ConvolveSteps<T=Vec<T>>{pub fn inv(&self,deg:usize)->Self{debug_assert!(!self[0].is_zero());let mut f=Self::from(T::one()/self[0].clone());let mut i=1;while i<deg{let g=self.prefix_ref((i*2).min(deg));let h=f.clone();let mut g=C::transform(g.data,2*i);let h=C::transform(h.data,2*i);C::multiply(&mut g,&h);let mut g=Self::from_vec(C::inverse_transform(g,2*i));g>>=i;let mut g=C::transform(g.data,2*i);C::multiply(&mut g,&h);let g=Self::from_vec(C::inverse_transform(g,2*i));f.data.extend((-g).into_iter().take(i));i*=2;}f.truncate(deg);f}pub fn exp(&self,deg:usize)->Self{debug_assert!(self[0].is_zero());let mut f=Self::one();let mut i=1;while i<deg{let mut g=-f.log(i*2);g[0]+=T::one();for(g,x)in g.iter_mut().zip(self.iter().take(i*2)){*g+=x.clone();}f=(f*g).prefix(i*2);i*=2;}f.prefix(deg)}pub fn log(&self,deg:usize)->Self{(self.inv(deg)*self.clone().diff()).integral().prefix(deg)}pub fn pow(&self,rhs:usize,deg:usize)->Self{if let Some(k)=self.iter().position(|x|!x.is_zero()){if k*rhs>=deg{Self::zeros(deg)}else{let mut x0=self[k].clone();let rev=T::one()/x0.clone();let x={let mut x=T::one();let mut y=rhs;while y>0{if y&1==1{x*=x0.clone();}x0*=x0.clone();y>>=1;}x};let mut f=(self.clone()*&rev)>>k;f=(f.log(deg)*&T::from(rhs)).exp(deg)*&x;f.truncate(deg-k*rhs);f<<=k*rhs;f}}else{Self::zeros(deg)}}}impl<T,C>FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficientSqrt,C:ConvolveSteps<T=Vec<T>>{pub fn sqrt(&self,deg:usize)->Option<Self>{if self[0].is_zero(){if let Some(k)=self.iter().position(|x|!x.is_zero()){if k%2!=0{return None;}else if deg>k/2{return Some((self>>k).sqrt(deg-k/2)?<<(k/2));}}}else{let inv2=T::one()/(T::one()+T::one());let mut f=Self::from(self[0].sqrt_coefficient()?);let mut i=1;while i<deg{f=(&f+&(self.prefix_ref(i*2)*f.inv(i*2))).prefix(i*2)*&inv2;i*=2;}f.truncate(deg);return Some(f);}Some(Self::zeros(deg))}}impl<T,C>FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient,C:ConvolveSteps<T=Vec<T>>{pub fn count_subset_sum<F>(&self,deg:usize,mut inverse:F)->Self where F:FnMut(usize)->T{let n=self.length();let mut f=Self::zeros(n);for i in 1..n{if!self[i].is_zero(){for(j,d)in(0..n).step_by(i).enumerate().skip(1){if j&1!=0{f[d]+=self[i].clone()*&inverse(j);}else{f[d]-=self[i].clone()*&inverse(j);}}}}f.exp(deg)}pub fn count_multiset_sum<F>(&self,deg:usize,mut inverse:F)->Self where F:FnMut(usize)->T{let n=self.length();let mut f=Self::zeros(n);for i in 1..n{if!self[i].is_zero(){for(j,d)in(0..n).step_by(i).enumerate().skip(1){f[d]+=self[i].clone()*&inverse(j);}}}f.exp(deg)}pub fn bostan_mori(self,rhs:Self,mut n:usize)->T{let mut p=self;let mut q=rhs;while n>0{let mut mq=q.clone();mq.iter_mut().skip(1).step_by(2).for_each(|x|*x=-x.clone());let u=p*mq.clone();p=if n%2==0{u.even()}else{u.odd()};q=(q*mq).even();n/=2;}p[0].clone()/q[0].clone()}fn middle_product(self,other:&C::F,deg:usize)->Self{let n=self.length();let mut s=C::transform(self.reversed().data,deg);C::multiply(&mut s,other);Self::from_vec((C::inverse_transform(s,deg))[n-1..].to_vec())}pub fn multipoint_evaluation(self,points:&[T])->Vec<T>{let n=points.len();if n<=32{return points.iter().map(|p|self.eval(p.clone())).collect();}let mut subproduct_tree=Vec::with_capacity(n*2);subproduct_tree.resize_with(n,Zero::zero);for x in points{subproduct_tree.push(Self::from_vec(vec![-x.clone(),T::one()]));}for i in(1..n).rev(){subproduct_tree[i]=&subproduct_tree[i*2]*&subproduct_tree[i*2+1];}let mut uptree_t=Vec::with_capacity(n*2);uptree_t.resize_with(1,Zero::zero);subproduct_tree.reverse();subproduct_tree.pop();let m=self.length();let v=subproduct_tree.pop().unwrap().reversed().resized(m);let s=C::transform(self.data,m*2);uptree_t.push(v.inv(m).middle_product(&s,m*2).resized(n).reversed());for i in 1..n{let subl=subproduct_tree.pop().unwrap();let subr=subproduct_tree.pop().unwrap();let(dl,dr)=(subl.length(),subr.length());let len=dl.max(dr)+uptree_t[i].length();let s=C::transform(uptree_t[i].data.to_vec(),len);uptree_t.push(subr.middle_product(&s,len).prefix(dl));uptree_t.push(subl.middle_product(&s,len).prefix(dr));}uptree_t[n..].iter().map(|u|u.data.get(0).cloned().unwrap_or_else(Zero::zero)).collect()}}impl<M,C>FormalPowerSeries<MInt<M>,C>where M:MIntConvert<usize>,C:ConvolveSteps<T=Vec<MInt<M>>>{#[doc=" f(x) <- f(x + a)"]pub fn taylor_shift(mut self,a:MInt<M>,f:&MemorizedFactorial<M>)->Self{let n=self.length();for i in 0..n{self.data[i]*=f.fact[i];}self.data.reverse();let mut b=a;let mut g=Self::from_vec(f.inv_fact[..n].to_vec());for i in 1..n{g[i]*=b;b*=a;}self*=g;self.truncate(n);self.data.reverse();for i in 0..n{self.data[i]*=f.inv_fact[i];}self}}}
mod formal_power_series_nums{#![allow(clippy::suspicious_arithmetic_impl,clippy::suspicious_op_assign_impl)]use super::*;use std::{mem::take,ops::{Add,AddAssign,Div,DivAssign,Mul,MulAssign,Neg,Rem,RemAssign,Shl,ShlAssign,Shr,ShrAssign,Sub,SubAssign}};impl<T,C>AddAssign<T>for FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient{fn add_assign(&mut self,rhs:T){if self.length()==0{self.data.push(T::zero());}self.data[0].add_assign(rhs);}}impl<T,C>SubAssign<T>for FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient{fn sub_assign(&mut self,rhs:T){if self.length()==0{self.data.push(T::zero());}self.data[0].sub_assign(rhs);self.trim_tail_zeros();}}impl<T,C>MulAssign<T>for FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient{fn mul_assign(&mut self,rhs:T){for x in self.iter_mut(){x.mul_assign(&rhs);}}}impl<T,C>DivAssign<T>for FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient{fn div_assign(&mut self,rhs:T){let rinv=T::one()/rhs;for x in self.iter_mut(){x.mul_assign(&rinv);}}}macro_rules!impl_fps_single_binop{($imp:ident,$method:ident,$imp_assign:ident,$method_assign:ident)=>{impl<T,C>$imp_assign<&T>for FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient,{fn$method_assign(&mut self,rhs:&T){$imp_assign::$method_assign(self,rhs.clone());}}impl<T,C>$imp<T>for FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient,{type Output=Self;fn$method(mut self,rhs:T)->Self::Output{$imp_assign::$method_assign(&mut self,rhs);self}}impl<T,C>$imp<&T>for FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient,{type Output=Self;fn$method(mut self,rhs:&T)->Self::Output{$imp_assign::$method_assign(&mut self,rhs);self}}impl<T,C>$imp<T>for&FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient,{type Output=FormalPowerSeries<T,C>;fn$method(self,rhs:T)->Self::Output{$imp::$method(self.clone(),rhs)}}impl<T,C>$imp<&T>for&FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient,{type Output=FormalPowerSeries<T,C>;fn$method(self,rhs:&T)->Self::Output{$imp::$method(self.clone(),rhs)}}};}impl_fps_single_binop!(Add,add,AddAssign,add_assign);impl_fps_single_binop!(Sub,sub,SubAssign,sub_assign);impl_fps_single_binop!(Mul,mul,MulAssign,mul_assign);impl_fps_single_binop!(Div,div,DivAssign,div_assign);impl<T,C>AddAssign<&Self>for FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient{fn add_assign(&mut self,rhs:&Self){if self.length()<rhs.length(){self.resize(rhs.length());}for(x,y)in self.iter_mut().zip(rhs.iter()){x.add_assign(y);}}}impl<T,C>SubAssign<&Self>for FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient{fn sub_assign(&mut self,rhs:&Self){if self.length()<rhs.length(){self.resize(rhs.length());}for(x,y)in self.iter_mut().zip(rhs.iter()){x.sub_assign(y);}self.trim_tail_zeros();}}macro_rules!impl_fps_binop_addsub{($imp:ident,$method:ident,$imp_assign:ident,$method_assign:ident)=>{impl<T,C>$imp_assign for FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient,{fn$method_assign(&mut self,rhs:Self){$imp_assign::$method_assign(self,&rhs);}}impl<T,C>$imp for FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient,{type Output=Self;fn$method(mut self,rhs:Self)->Self::Output{$imp_assign::$method_assign(&mut self,&rhs);self}}impl<T,C>$imp<&FormalPowerSeries<T,C>>for FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient,{type Output=Self;fn$method(mut self,rhs:&FormalPowerSeries<T,C>)->Self::Output{$imp_assign::$method_assign(&mut self,rhs);self}}impl<T,C>$imp<FormalPowerSeries<T,C>>for&FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient,{type Output=FormalPowerSeries<T,C>;fn$method(self,rhs:FormalPowerSeries<T,C>)->Self::Output{let mut self_=self.clone();$imp_assign::$method_assign(&mut self_,&rhs);self_}}impl<T,C>$imp<&FormalPowerSeries<T,C>>for&FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient,{type Output=FormalPowerSeries<T,C>;fn$method(self,rhs:&FormalPowerSeries<T,C>)->Self::Output{let mut self_=self.clone();$imp_assign::$method_assign(&mut self_,rhs);self_}}};}impl_fps_binop_addsub!(Add,add,AddAssign,add_assign);impl_fps_binop_addsub!(Sub,sub,SubAssign,sub_assign);impl<T,C>Mul for FormalPowerSeries<T,C>where C:ConvolveSteps<T=Vec<T>>{type Output=Self;fn mul(self,rhs:Self)->Self::Output{Self::from_vec(C::convolve(self.data,rhs.data))}}impl<T,C>Div for FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient,C:ConvolveSteps<T=Vec<T>>{type Output=Self;fn div(mut self,mut rhs:Self)->Self::Output{self.trim_tail_zeros();rhs.trim_tail_zeros();if self.length()<rhs.length(){return Self::zero();}self.data.reverse();rhs.data.reverse();let n=self.length()-rhs.length()+1;let mut res=self*rhs.inv(n);res.truncate(n);res.data.reverse();res}}impl<T,C>Rem for FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient,C:ConvolveSteps<T=Vec<T>>{type Output=Self;fn rem(self,rhs:Self)->Self::Output{let mut rem=self.clone()-self/rhs.clone()*rhs;rem.trim_tail_zeros();rem}}impl<T,C>FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient,C:ConvolveSteps<T=Vec<T>>{pub fn div_rem(self,rhs:Self)->(Self,Self){let div=self.clone()/rhs.clone();let mut rem=self-div.clone()*rhs;rem.trim_tail_zeros();(div,rem)}}macro_rules!impl_fps_binop_conv{($imp:ident,$method:ident,$imp_assign:ident,$method_assign:ident)=>{impl<T,C>$imp_assign for FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient,C:ConvolveSteps<T=Vec<T>>,{fn$method_assign(&mut self,rhs:Self){*self=$imp::$method(Self::from_vec(take(&mut self.data)),rhs);}}impl<T,C>$imp_assign<&Self>for FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient,C:ConvolveSteps<T=Vec<T>>,{fn$method_assign(&mut self,rhs:&Self){$imp_assign::$method_assign(self,rhs.clone());}}impl<T,C>$imp<&FormalPowerSeries<T,C>>for FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient,C:ConvolveSteps<T=Vec<T>>,{type Output=Self;fn$method(self,rhs:&FormalPowerSeries<T,C>)->Self::Output{$imp::$method(self,rhs.clone())}}impl<T,C>$imp<FormalPowerSeries<T,C>>for&FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient,C:ConvolveSteps<T=Vec<T>>,{type Output=FormalPowerSeries<T,C>;fn$method(self,rhs:FormalPowerSeries<T,C>)->Self::Output{$imp::$method(self.clone(),rhs)}}impl<T,C>$imp<&FormalPowerSeries<T,C>>for&FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient,C:ConvolveSteps<T=Vec<T>>,{type Output=FormalPowerSeries<T,C>;fn$method(self,rhs:&FormalPowerSeries<T,C>)->Self::Output{$imp::$method(self.clone(),rhs.clone())}}};}impl_fps_binop_conv!(Mul,mul,MulAssign,mul_assign);impl_fps_binop_conv!(Div,div,DivAssign,div_assign);impl_fps_binop_conv!(Rem,rem,RemAssign,rem_assign);impl<T,C>Neg for FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient{type Output=Self;fn neg(mut self)->Self::Output{for x in self.iter_mut(){*x=-x.clone();}self}}impl<T,C>Neg for&FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient{type Output=FormalPowerSeries<T,C>;fn neg(self)->Self::Output{self.clone().neg()}}impl<T,C>ShrAssign<usize>for FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient{fn shr_assign(&mut self,rhs:usize){if self.length()<=rhs{*self=Self::zero();}else{for i in rhs..self.length(){self[i-rhs]=self[i].clone();}self.truncate(self.length()-rhs);}}}impl<T,C>ShlAssign<usize>for FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient{fn shl_assign(&mut self,rhs:usize){let n=self.length();self.resize(n+rhs);for i in(0..n).rev(){self[i+rhs]=self[i].clone();}for i in 0..rhs{self[i]=T::zero();}}}impl<T,C>Shr<usize>for FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient{type Output=Self;fn shr(mut self,rhs:usize)->Self::Output{self.shr_assign(rhs);self}}impl<T,C>Shl<usize>for FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient{type Output=Self;fn shl(mut self,rhs:usize)->Self::Output{self.shl_assign(rhs);self}}impl<T,C>Shr<usize>for&FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient{type Output=FormalPowerSeries<T,C>;fn shr(self,rhs:usize)->Self::Output{if self.length()<=rhs{Self::Output::zero()}else{let mut f=Self::Output::zeros(self.length()-rhs);for i in rhs..self.length(){f[i-rhs]=self[i].clone();}f}}}impl<T,C>Shl<usize>for&FormalPowerSeries<T,C>where T:FormalPowerSeriesCoefficient{type Output=FormalPowerSeries<T,C>;fn shl(self,rhs:usize)->Self::Output{let mut f=Self::Output::zeros(self.length()+rhs);for(i,x)in self.iter().cloned().enumerate().rev(){f[i+rhs]=x;}f}}}
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;fn local_key()->&'static std::thread::LocalKey<std::cell::UnsafeCell<Self::T>>;#[inline]fn get()->Self::T{Self::with(Clone::clone)}#[inline]fn set(x:Self::T){Self::local_key().with(|cell|unsafe{*cell.get()=x})}#[inline]fn with<F,R>(f:F)->R where F:FnOnce(&Self::T)->R{Self::local_key().with(|cell|unsafe{f(&*cell.get())})}#[inline]fn modify<F,R>(f:F)->R where F:FnOnce(&mut Self::T)->R{Self::local_key().with(|cell|unsafe{f(&mut*cell.get())})}}#[doc=" Implement [`AssociatedValue`]."]#[doc=""]#[doc=" [`AssociatedValue`]: super::AssociatedValue"]#[doc=""]#[doc=" # Examples"]#[doc=""]#[doc=" ```"]#[doc=" use competitive::tools::AssociatedValue;"]#[doc=" struct 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=" struct 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=$t;#[inline]fn local_key()->&'static::std::thread::LocalKey<::std::cell::UnsafeCell<Self::T>>{::std::thread_local!(static __LOCAL_KEY: ::std::cell::UnsafeCell<$t> =::std::cell::UnsafeCell::new($e));&__LOCAL_KEY}}};}}
pub mod mint_basic{use super::*;#[macro_export]macro_rules!define_basic_mintbase{($name:ident,$m:expr,$basety:ty,$signedty:ty,$upperty:ty,[$($unsigned:ty),*],[$($signed:ty),*])=>{pub struct$name;impl MIntBase for$name{type Inner=$basety;#[inline]fn get_mod()->Self::Inner{$m}#[inline]fn mod_zero()->Self::Inner{0}#[inline]fn mod_one()->Self::Inner{1}#[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{(x as$upperty*y as$upperty%Self::get_mod()as$upperty)as$basety}#[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$signedty;let(mut a,mut b)=(x as$signedty,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);}(if x<0{x+p}else{x})as _}}$(impl MIntConvert<$unsigned>for$name{#[inline]fn from(x:$unsigned)->Self::Inner{(x%<Self as MIntBase>::get_mod()as$unsigned)as$basety}#[inline]fn into(x:Self::Inner)->$unsigned{x as$unsigned}#[inline]fn mod_into()->$unsigned{<Self as MIntBase>::get_mod()as$unsigned}})*$(impl MIntConvert<$signed>for$name{#[inline]fn from(x:$signed)->Self::Inner{let x=x%<Self as MIntBase>::get_mod()as$signed;if x<0{(x+<Self as MIntBase>::get_mod()as$signed)as$basety}else{x as$basety}}#[inline]fn into(x:Self::Inner)->$signed{x as$signed}#[inline]fn mod_into()->$signed{<Self as MIntBase>::get_mod()as$signed}})*};}#[macro_export]macro_rules!define_basic_mint32{($([$name:ident,$m:expr,$mint_name:ident]),*)=>{$(crate::define_basic_mintbase!($name,$m,u32,i32,u64,[u32,u64,u128,usize],[i32,i64,i128,isize]);pub type$mint_name=MInt<$name>;)*};}define_basic_mint32!([Modulo998244353,998_244_353,MInt998244353],[Modulo1000000007,1_000_000_007,MInt1000000007],[Modulo1000000009,1_000_000_009,MInt1000000009],[DynModuloU32,DYN_MODULUS_U32.with(|cell|unsafe{*cell.get()}),DynMIntU32]);thread_local!(static DYN_MODULUS_U32:std::cell::UnsafeCell<u32> =std::cell::UnsafeCell::new(1_000_000_007));impl DynModuloU32{pub fn set_mod(m:u32){DYN_MODULUS_U32.with(|cell|unsafe{*cell.get()=m})}}thread_local!(static DYN_MODULUS_U64:std::cell::UnsafeCell<u64> =std::cell::UnsafeCell::new(1_000_000_007));define_basic_mintbase!(DynModuloU64,DYN_MODULUS_U64.with(|cell|unsafe{*cell.get()}),u64,i64,u128,[u64,u128,usize],[i64,i128,isize]);impl DynModuloU64{pub fn set_mod(m:u64){DYN_MODULUS_U64.with(|cell|unsafe{*cell.get()=m})}}pub type DynMIntU64=MInt<DynModuloU64>;pub struct Modulo2;impl MIntBase for Modulo2{type Inner=u32;#[inline]fn get_mod()->Self::Inner{2}#[inline]fn mod_zero()->Self::Inner{0}#[inline]fn mod_one()->Self::Inner{1}#[inline]fn mod_add(x:Self::Inner,y:Self::Inner)->Self::Inner{x^y}#[inline]fn mod_sub(x:Self::Inner,y:Self::Inner)->Self::Inner{x^y}#[inline]fn mod_mul(x:Self::Inner,y:Self::Inner)->Self::Inner{x&y}#[inline]fn mod_div(x:Self::Inner,y:Self::Inner)->Self::Inner{assert_ne!(y,0);x}#[inline]fn mod_neg(x:Self::Inner)->Self::Inner{x}#[inline]fn mod_inv(x:Self::Inner)->Self::Inner{assert_ne!(x,0);x}#[inline]fn mod_pow(x:Self::Inner,y:usize)->Self::Inner{if y==0{1}else{x}}}macro_rules!impl_to_mint_base_for_modulo2{($name:ident,$basety:ty,[$($t:ty),*])=>{$(impl MIntConvert<$t>for$name{#[inline]fn from(x:$t)->Self::Inner{(x&1)as$basety}#[inline]fn into(x:Self::Inner)->$t{x as$t}#[inline]fn mod_into()->$t{1}})*};}impl_to_mint_base_for_modulo2!(Modulo2,u32,[u8,u16,u32,u64,u128,usize,i8,i16,i32,i64,i128,isize]);pub type MInt2=MInt<Modulo2>;}
#[repr(transparent)]pub struct MInt<M>where M:MIntBase{x:M::Inner,_marker:std::marker::PhantomData<fn()->M>}
pub trait MIntBase{type Inner:Sized+Copy+Eq+std::fmt::Debug+std::hash::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}}
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;}
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};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::$method(self,*other);}}};}impl_mint_ref_op_assign!(AddAssign,add_assign,MInt<M>,Add,add);impl_mint_ref_op_assign!(SubAssign,sub_assign,MInt<M>,Sub,sub);impl_mint_ref_op_assign!(MulAssign,mul_assign,MInt<M>,Mul,mul);impl_mint_ref_op_assign!(DivAssign,div_assign,MInt<M>,Div,div);}
use std::marker::PhantomData;
pub struct NumberTheoreticTransform<M>(PhantomData<fn()->M>);
pub trait NttModulus:'static+Sized+MIntBase<Inner=u32>+MIntConvert<u32>+AssociatedValue<T=number_theoretic_transform_impls::NttCache<Self>>{fn primitive_root()->MInt<Self>;}
pub struct Convolve<M>(PhantomData<fn()->M>);
pub type Convolve998244353=Convolve<mint_basic::Modulo998244353>;
pub type MIntConvolve<M>=Convolve<(M,(number_theoretic_transform_impls::Modulo2013265921,number_theoretic_transform_impls::Modulo1811939329,number_theoretic_transform_impls::Modulo2113929217))>;
pub trait ConvolveSteps{type T;type F;fn length(t:&Self::T)->usize;fn transform(t:Self::T,len:usize)->Self::F;fn inverse_transform(f:Self::F,len:usize)->Self::T;fn multiply(f:&mut Self::F,g:&Self::F);fn convolve(a:Self::T,b:Self::T)->Self::T{let len=(Self::length(&a)+Self::length(&b)).saturating_sub(1);let mut a=Self::transform(a,len);let b=Self::transform(b,len);Self::multiply(&mut a,&b);Self::inverse_transform(a,len)}}
pub mod number_theoretic_transform_impls{use super::*;use mint_basic::Modulo998244353;macro_rules!impl_ntt_modulus{($([$name:ident,$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]);crate::define_basic_mint32!([Modulo2113929217,2_113_929_217,MInt2113929217],[Modulo1811939329,1_811_939_329,MInt1811939329],[Modulo2013265921,2_013_265_921,MInt2013265921]);#[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>NumberTheoreticTransform<M>where M:NttModulus{#[allow(clippy::needless_range_loop)]fn ntt(a:&mut[MInt<M>]){M::modify(|cache|{let n=a.len();cache.ensure(n/2);let mut u=1;let mut v=n/2;for i in(1..=n.trailing_zeros()).rev(){for jh in 0..u{let wj=cache.cache[jh];let p=jh<<i;let(l,r)=a.split_at_mut(p+v);for(x,y)in l[p..].iter_mut().zip(&mut r[..v]){let ajv=wj**y;*y=*x-ajv;*x+=ajv;}}u<<=1;v>>=1;}});}#[allow(clippy::needless_range_loop)]fn intt(a:&mut[MInt<M>]){M::modify(|cache|{let n=a.len();cache.ensure(n/2);let mut u=n/2;let mut v=1;for i in 1..=n.trailing_zeros(){for jh in 0..u{let wj=cache.icache[jh];let p=jh<<i;let(l,r)=a.split_at_mut(p+v);for(x,y)in l[p..].iter_mut().zip(&mut r[..v]){let ajv=*x-*y;*x+=*y;*y=wj*ajv;}}u>>=1;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);NumberTheoreticTransform::<M>::ntt(&mut t);t}fn inverse_transform(mut f:Self::F,len:usize)->Self::T{NumberTheoreticTransform::<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);NumberTheoreticTransform::<N1>::ntt(&mut f.0);NumberTheoreticTransform::<N2>::ntt(&mut f.1);NumberTheoreticTransform::<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;}}}}
#[derive(Clone,Debug)]pub struct MemorizedFactorial<M:MIntConvert<usize>>{pub fact:Vec<MInt<M>>,pub inv_fact:Vec<MInt<M>>}
impl<M:MIntConvert<usize>>MemorizedFactorial<M>{pub fn new(max_n:usize)->Self{let mut fact=vec![MInt::one();max_n+1];let mut inv_fact=vec![MInt::one();max_n+1];for i in 2..=max_n{fact[i]=fact[i-1]*MInt::from(i);}inv_fact[max_n]=fact[max_n].inv();for i in(3..=max_n).rev(){inv_fact[i-1]=inv_fact[i]*MInt::from(i);}Self{fact,inv_fact}}#[inline]pub fn combination(&self,n:usize,r:usize)->MInt<M>{debug_assert!(n<self.fact.len());if r<=n{self.fact[n]*self.inv_fact[r]*self.inv_fact[n-r]}else{MInt::zero()}}#[inline]pub fn permutation(&self,n:usize,r:usize)->MInt<M>{debug_assert!(n<self.fact.len());if r<=n{self.fact[n]*self.inv_fact[n-r]}else{MInt::zero()}}#[inline]pub fn homogeneous_product(&self,n:usize,r:usize)->MInt<M>{debug_assert!(n+r<self.fact.len()+1);if n!=0&&r!=0{self.combination(n+r-1,r)}else{MInt::one()}}#[inline]pub fn inv(&self,n:usize)->MInt<M>{debug_assert!(n<self.fact.len());debug_assert!(n>0);self.inv_fact[n]*self.fact[n-1]}}
impl<M:MIntConvert<u32>>MInt<M>{pub fn sqrt(self)->Option<Self>{fn jacobi<M:MIntConvert<u32>>(mut x:u32)->i8{let mut s=1i8;let mut m=M::mod_into();while m>1{x%=m;if x==0{return 0;}let k=x.trailing_zeros();if k%2==1&&(m+2)&4!=0{s=-s;}x>>=k;if x&m&2!=0{s=-s;}std::mem::swap(&mut x,&mut m);}s}if M::mod_into()==2{return Some(self);}let j=jacobi::<M>(u32::from(self));match j.cmp(&0){std::cmp::Ordering::Less=>{return None;}std::cmp::Ordering::Equal=>{return Some(Self::zero());}std::cmp::Ordering::Greater=>{}}let mut r=1;let(mut f0,d)=loop{r^=r<<5;r^=r>>17;r^=r<<11;let b=Self::from(r);let d=b*b-self;if jacobi::<M>(u32::from(d))==-1{break(b,d);}};let(mut f1,mut g0,mut g1,mut e)=(Self::one(),Self::one(),Self::zero(),(M::mod_into()+1)/2);while e>0{if e%2==1{let t=g0*f0+d*g1*f1;g1=g0*f1+g1*f0;g0=t;}let t=f0*f0+d*f1*f1;f1=Self::from(2)*f0*f1;f0=t;e/=2;}if u32::from(g0)>M::mod_into()-u32::from(g0){g0=-g0;}Some(g0)}}
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.});}