pub fn solve() {
    crate::prepare!();
    sc!(n, bc: [(Usize1, usize); n]);
    let mut xs = vec![];
    for &(b, c) in &bc {
        xs.push(b);
        xs.push(c);
    }
    xs.sort_unstable();
    xs.dedup();
    let m = xs.len();
    let idx: HashMap<_, _> = xs.iter().enumerate().map(|(i, &x)| (x, i)).collect();
    let mut seg = LazySegmentTree::<RangeSumRangeAdd<M>>::from_vec({
        let mut v = vec![(M::zero(), M::zero()); m - 1];
        for (i, w) in xs.windows(2).enumerate() {
            v[i].1 = M::from(w[1] - w[0]);
        }
        v
    });
    let mut ans = M::from(n * (n - 1) / 2);
    for (b, c) in bc {
        let (l, r) = (idx[&b], idx[&c]);
        let d = M::from(c - b).inv();
        ans -= seg.fold(l, r).0 * d;
        seg.update(l, r, d);
    }
    for i in 3..=n {
        ans *= M::from(i);
    }
    pp!(ans);
}
pub type M = mint_basic::MInt998244353;
crate::main!();
#[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)*)}}};()=>{$crate::prepare!(@output($));$crate::prepare!(@normal($))};(?)=>{$crate::prepare!(@output($));$crate::prepare!(@interactive($))};}#[macro_export]macro_rules!main{()=>{fn main(){solve();}};(avx2)=>{fn main(){#[target_feature(enable="avx2")]unsafe fn solve_avx2(){solve();}unsafe{solve_avx2()}}};(large_stack)=>{fn main(){const STACK_SIZE:usize=512*1024*1024;::std::thread::Builder::new().stack_size(STACK_SIZE).spawn(solve).unwrap().join().unwrap();}};}}
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=" - `@it1 $expr`: print iterator as 1-indexed"]#[doc=" - `@cw ($char $expr)`: print iterator as `(elem as u8 + $char as u8) as char`"]#[doc=" - `@bw ($byte $expr)`: print iterator as `(elem as u8 + $byte) as char`"]#[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);}}};(@@it1$writer:expr,$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,$is_head,item+1);}for item in iter{$crate::iter_print!(@@item$writer,$sep,false,item+1);}}};(@@cw$writer:expr,$sep:expr,$is_head:expr,($ch:literal$iter:expr))=>{{let mut iter=$iter.into_iter();let b=$ch as u8;if let Some(item)=iter.next(){$crate::iter_print!(@@item$writer,$sep,$is_head,(item as u8+b)as char);}for item in iter{$crate::iter_print!(@@item$writer,$sep,false,(item as u8+b)as char);}}};(@@bw$writer:expr,$sep:expr,$is_head:expr,($b:literal$iter:expr))=>{{let mut iter=$iter.into_iter();let b:u8=$b;if let Some(item)=iter.next(){$crate::iter_print!(@@item$writer,$sep,$is_head,(item as u8+b)as char);}for item in iter{$crate::iter_print!(@@item$writer,$sep,false,(item as u8+b)as char);}}};(@@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 it1)=>{};(@@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,$is_head:expr,@fmt$arg:tt$($t:tt)*)=>{$crate::iter_print!(@@fmt$writer,$sep,$is_head,$arg);$crate::iter_print!(@@inner$writer,$sep,$is_head,$($t)*);};(@@inner$writer:expr,$sep:expr,$is_head:expr,@cw$arg:tt$($t:tt)*)=>{$crate::iter_print!(@@cw$writer,$sep,$is_head,$arg);$crate::iter_print!(@@inner$writer,$sep,$is_head,$($t)*);};(@@inner$writer:expr,$sep:expr,$is_head:expr,@bw$arg:tt$($t:tt)*)=>{$crate::iter_print!(@@bw$writer,$sep,$is_head,$arg);$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]]};[|$i:pat_param|$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::counter::{BTreeCounter,HashCounter};
mod counter{use std::{borrow::Borrow,collections::{btree_map,hash_map,BTreeMap,HashMap},fmt::{self,Debug},hash::Hash,iter::{Extend,FromIterator},ops::RangeBounds};#[derive(Clone)]pub struct HashCounter<T>{map:HashMap<T,usize>}impl<T>Debug for HashCounter<T>where T:Debug{fn fmt(&self,f:&mut fmt::Formatter<'_>)->fmt::Result{f.debug_map().entries(self.iter()).finish()}}impl<T>Default for HashCounter<T>where T:Eq+Hash{fn default()->Self{Self{map:HashMap::default()}}}impl<T>HashCounter<T>{pub fn len(&self)->usize{self.map.len()}pub fn is_empty(&self)->bool{self.map.is_empty()}pub fn clear(&mut self){self.map.clear();}pub fn keys(&self)->hash_map::Keys<'_,T,usize>{self.map.keys()}pub fn values(&self)->hash_map::Values<'_,T,usize>{self.map.values()}pub fn iter(&self)->hash_map::Iter<'_,T,usize>{self.map.iter()}pub fn drain(&mut self)->hash_map::Drain<'_,T,usize>{self.map.drain()}}impl<T>HashCounter<T>where T:Eq+Hash{pub fn new()->Self{Self::default()}pub fn with_capacity(capacity:usize)->Self{Self{map:HashMap::with_capacity(capacity)}}pub fn get(&self,item:&T)->usize{self.map.get(item).cloned().unwrap_or_default()}pub fn add(&mut self,item:T){self.add_count(item,1)}pub fn add_count(&mut self,item:T,count:usize){*self.map.entry(item).or_default()+=count;}pub fn remove(&mut self,item:&T)->bool{self.remove_count(item,1)==1}pub fn remove_count(&mut self,item:&T,count:usize)->usize{if let Some(cnt)=self.map.get_mut(item){if*cnt<=count{let cnt=*cnt;self.map.remove(item);cnt}else{*cnt-=count;count}}else{0}}pub fn append(&mut self,other:&mut Self){if self.map.len()<other.map.len(){std::mem::swap(self,other);}self.extend(other.map.drain());}}impl<T>Extend<T>for HashCounter<T>where T:Eq+Hash{fn extend<I:IntoIterator<Item=T>>(&mut self,iter:I){for item in iter{self.add(item)}}}impl<T>Extend<(T,usize)>for HashCounter<T>where T:Eq+Hash{fn extend<I:IntoIterator<Item=(T,usize)>>(&mut self,iter:I){for(item,count)in iter{self.add_count(item,count)}}}impl<T>FromIterator<T>for HashCounter<T>where T:Eq+Hash{fn from_iter<I:IntoIterator<Item=T>>(iter:I)->Self{let mut map=Self::default();map.extend(iter);map}}impl<T>FromIterator<(T,usize)>for HashCounter<T>where T:Eq+Hash{fn from_iter<I:IntoIterator<Item=(T,usize)>>(iter:I)->Self{let mut map=Self::default();map.extend(iter);map}}#[derive(Clone)]pub struct BTreeCounter<T>{map:BTreeMap<T,usize>}impl<T>Debug for BTreeCounter<T>where T:Debug{fn fmt(&self,f:&mut fmt::Formatter<'_>)->fmt::Result{f.debug_map().entries(self.iter()).finish()}}impl<T>Default for BTreeCounter<T>where T:Ord{fn default()->Self{Self{map:BTreeMap::default()}}}impl<T>BTreeCounter<T>{pub fn len(&self)->usize{self.map.len()}pub fn is_empty(&self)->bool{self.map.is_empty()}pub fn keys(&self)->btree_map::Keys<'_,T,usize>{self.map.keys()}pub fn values(&self)->btree_map::Values<'_,T,usize>{self.map.values()}pub fn iter(&self)->btree_map::Iter<'_,T,usize>{self.map.iter()}pub fn range<Q,R>(&self,range:R)->btree_map::Range<'_,T,usize>where Q:Ord,R:RangeBounds<Q>,T:Borrow<Q>+Ord{self.map.range(range)}}impl<T>BTreeCounter<T>where T:Ord{pub fn new()->Self{Self::default()}pub fn clear(&mut self){self.map.clear();}pub fn get(&self,item:&T)->usize{self.map.get(item).cloned().unwrap_or_default()}pub fn add(&mut self,item:T){self.add_count(item,1)}pub fn add_count(&mut self,item:T,count:usize){*self.map.entry(item).or_default()+=count;}pub fn remove(&mut self,item:&T)->bool{self.remove_count(item,1)==1}pub fn remove_count(&mut self,item:&T,count:usize)->usize{if let Some(cnt)=self.map.get_mut(item){if*cnt<=count{let cnt=*cnt;self.map.remove(item);cnt}else{*cnt-=count;count}}else{0}}}impl<T>Extend<T>for BTreeCounter<T>where T:Ord{fn extend<I:IntoIterator<Item=T>>(&mut self,iter:I){for item in iter{self.add(item)}}}impl<T>Extend<(T,usize)>for BTreeCounter<T>where T:Ord{fn extend<I:IntoIterator<Item=(T,usize)>>(&mut self,iter:I){for(item,count)in iter{self.add_count(item,count)}}}impl<T>FromIterator<T>for BTreeCounter<T>where T:Ord{fn from_iter<I:IntoIterator<Item=T>>(iter:I)->Self{let mut map=Self::default();map.extend(iter);map}}impl<T>FromIterator<(T,usize)>for BTreeCounter<T>where T:Ord{fn from_iter<I:IntoIterator<Item=(T,usize)>>(iter:I)->Self{let mut map=Self::default();map.extend(iter);map}}}
pub mod mint_basic{use super::*;use std::{cell::UnsafeCell,mem::swap};#[macro_export]macro_rules!define_basic_mintbase{($name:ident,$m:expr,$basety:ty,$signedty:ty,$upperty:ty,[$($unsigned:ty),*],[$($signed:ty),*])=>{pub enum$name{}impl MIntBase for$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{$name::rem(x as$upperty*y 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;swap(&mut x,&mut u);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]),*)=>{$(define_basic_mintbase!($name,$m,u32,i32,u64,[u32,u64,u128,usize],[i32,i64,i128,isize]);impl$name{fn rem(x:u64)->u64{x%$m}}pub type$mint_name=MInt<$name>;)*};}thread_local!(static DYN_MODULUS_U32:UnsafeCell<BarrettReduction<u64>> =UnsafeCell::new(BarrettReduction::<u64>::new(1_000_000_007)));impl DynModuloU32{pub fn set_mod(m:u32){DYN_MODULUS_U32.with(|cell|unsafe{*cell.get()=BarrettReduction::<u64>::new(m as u64)});}fn rem(x:u64)->u64{DYN_MODULUS_U32.with(|cell|unsafe{(*cell.get()).rem(x)})}}impl DynMIntU32{pub fn set_mod(m:u32){DynModuloU32::set_mod(m)}}thread_local!(static DYN_MODULUS_U64:UnsafeCell<BarrettReduction<u128>> =UnsafeCell::new(BarrettReduction::<u128>::new(1_000_000_007)));impl DynModuloU64{pub fn set_mod(m:u64){DYN_MODULUS_U64.with(|cell|unsafe{*cell.get()=BarrettReduction::<u128>::new(m as u128)})}fn rem(x:u128)->u128{DYN_MODULUS_U64.with(|cell|unsafe{(*cell.get()).rem(x)})}}impl DynMIntU64{pub fn set_mod(m:u64){DynModuloU64::set_mod(m)}}define_basic_mint32!([Modulo998244353,998_244_353,MInt998244353],[Modulo1000000007,1_000_000_007,MInt1000000007],[Modulo1000000009,1_000_000_009,MInt1000000009]);define_basic_mintbase!(DynModuloU32,DYN_MODULUS_U32.with(|cell|unsafe{(*cell.get()).get_mod()as u32}),u32,i32,u64,[u32,u64,u128,usize],[i32,i64,i128,isize]);pub type DynMIntU32=MInt<DynModuloU32>;define_basic_mintbase!(DynModuloU64,DYN_MODULUS_U64.with(|cell|unsafe{(*cell.get()).get_mod()as u64}),u64,i64,u128,[u64,u128,usize],[i64,i128,isize]);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()->$t{1}})*};}impl_to_mint_base_for_modulo2!(Modulo2,u32,[u8,u16,u32,u64,u128,usize,i8,i16,i32,i64,i128,isize]);pub type MInt2=MInt<Modulo2>;}
pub use self::barrett_reduction::BarrettReduction;
mod barrett_reduction{#[derive(Debug,Clone,Copy)]pub struct BarrettReduction<T>{m:T,im:T}macro_rules!impl_barrett{($basety:ty,|$a:ident,$im:ident|$quotient:expr)=>{impl BarrettReduction<$basety>{pub fn new(m:$basety)->Self{Self{m,im:!0/m}}pub fn get_mod(&self)->$basety{self.m}pub fn div_rem(&self,$a:$basety)->($basety,$basety){if self.m==1{return($a,0);}let$im=self.im;let mut q=$quotient;let mut r=$a-q*self.m;if self.m<=r{r-=self.m;q+=1;}(q,r)}pub fn div(&self,a:$basety)->$basety{self.div_rem(a).0}pub fn rem(&self,a:$basety)->$basety{self.div_rem(a).1}}};}impl_barrett!(u32,|a,im|((a as u64*im as u64)>>32)as u32);impl_barrett!(u64,|a,im|((a as u128*im as u128)>>64)as u64);impl_barrett!(u128,|a,im|{const MASK64:u128=0xffff_ffff_ffff_ffff;let au=a>>64;let ad=a&MASK64;let imu=im>>64;let imd=im&MASK64;let mut res=au*imu;let x=(ad*imd)>>64;let(x,c)=x.overflowing_add(au*imd);res+=c as u128;let(x,c)=x.overflowing_add(ad*imu);res+=c as u128;res+(x>>64)});}
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 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::lazy_segment_tree::LazySegmentTree;
mod lazy_segment_tree{use super::MonoidAction;use std::{fmt::{self,Debug,Formatter},mem::replace};pub struct LazySegmentTree<M>where M:MonoidAction{n:usize,seg:Vec<(M::Agg,M::Act)>}impl<M>Clone for LazySegmentTree<M>where M:MonoidAction{fn clone(&self)->Self{Self{n:self.n,seg:self.seg.clone()}}}impl<M>Debug for LazySegmentTree<M>where M:MonoidAction,M::Agg:Debug,M::Act:Debug{fn fmt(&self,f:&mut Formatter<'_>)->fmt::Result{f.debug_struct("LazySegmentTree").field("n",&self.n).field("seg",&self.seg).finish()}}impl<M>LazySegmentTree<M>where M:MonoidAction{pub fn new(n:usize)->Self{let seg=vec![(M::agg_unit(),M::act_unit());2*n];Self{n,seg}}pub fn from_vec(v:Vec<M::Agg>)->Self{let n=v.len();let mut seg=vec![(M::agg_unit(),M::act_unit());2*n];for(i,x)in v.into_iter().enumerate(){seg[i+n].0=x;}for i in(1..n).rev(){seg[i].0=M::agg_operate(&seg[2*i].0,&seg[2*i+1].0);}Self{n,seg}}#[inline]fn update_at(&mut self,k:usize,x:&M::Act){let nx=M::act_agg(&self.seg[k].0,x);if k<self.n{self.seg[k].1=M::act_operate(&self.seg[k].1,x);}if let Some(nx)=nx{self.seg[k].0=nx;}else if k<self.n{self.propagate_at(k);self.recalc_at(k);}else{panic!("act failed on leaf");}}#[inline]fn recalc_at(&mut self,k:usize){self.seg[k].0=M::agg_operate(&self.seg[2*k].0,&self.seg[2*k+1].0);}#[inline]fn propagate_at(&mut self,k:usize){debug_assert!(k<self.n);let x=replace(&mut self.seg[k].1,M::act_unit());self.update_at(2*k,&x);self.update_at(2*k+1,&x);}#[inline]fn propagate(&mut self,k:usize,right:bool,nofilt:bool){let right=right as usize;for i in(1..(k+1-right).next_power_of_two().trailing_zeros()).rev(){if nofilt||(k>>i)<<i!=k{self.propagate_at((k-right)>>i);}}}#[inline]fn recalc(&mut self,k:usize,right:bool,nofilt:bool){let right=right as usize;for i in 1..(k+1-right).next_power_of_two().trailing_zeros(){if nofilt||(k>>i)<<i!=k{self.recalc_at((k-right)>>i);}}}pub fn update(&mut self,l:usize,r:usize,x:M::Act){debug_assert!(l<=r);debug_assert!(r<=self.n);let mut a=l+self.n;let mut b=r+self.n;self.propagate(a,false,false);self.propagate(b,true,false);while a<b{if a&1!=0{self.update_at(a,&x);a+=1;}if b&1!=0{b-=1;self.update_at(b,&x);}a/=2;b/=2;}self.recalc(l+self.n,false,false);self.recalc(r+self.n,true,false);}pub fn fold(&mut self,l:usize,r:usize)->M::Agg{debug_assert!(l<=r);debug_assert!(r<=self.n);let mut l=l+self.n;let mut r=r+self.n;self.propagate(l,false,true);self.propagate(r,true,true);let mut vl=M::agg_unit();let mut vr=M::agg_unit();while l<r{if l&1!=0{vl=M::agg_operate(&vl,&self.seg[l].0);l+=1;}if r&1!=0{r-=1;vr=M::agg_operate(&self.seg[r].0,&vr);}l/=2;r/=2;}M::agg_operate(&vl,&vr)}pub fn set(&mut self,k:usize,x:M::Agg){let k=k+self.n;self.propagate(k,false,true);self.seg[k]=(x,M::act_unit());self.recalc(k,false,true);}pub fn get(&mut self,k:usize)->M::Agg{self.fold(k,k+1)}pub fn fold_all(&mut self)->M::Agg{self.fold(0,self.n)}fn bisect_perfect<P>(&mut self,mut pos:usize,mut acc:M::Agg,p:P)->(usize,M::Agg)where P:Fn(&M::Agg)->bool{while pos<self.n{self.propagate_at(pos);pos<<=1;let nacc=M::agg_operate(&acc,&self.seg[pos].0);if!p(&nacc){acc=nacc;pos+=1;}}(pos-self.n,acc)}fn rbisect_perfect<P>(&mut self,mut pos:usize,mut acc:M::Agg,p:P)->(usize,M::Agg)where P:Fn(&M::Agg)->bool{while pos<self.n{self.propagate_at(pos);pos=pos*2+1;let nacc=M::agg_operate(&self.seg[pos].0,&acc);if!p(&nacc){acc=nacc;pos-=1;}}(pos-self.n,acc)}#[doc=" Returns the first index that satisfies a accumlative predicate."]pub fn position_acc<P>(&mut self,l:usize,r:usize,p:P)->Option<usize>where P:Fn(&M::Agg)->bool{let mut l=l+self.n;let r=r+self.n;self.propagate(l,false,true);self.propagate(r,true,true);let mut k=0usize;let mut acc=M::agg_unit();while l<r>>k{if l&1!=0{let nacc=M::agg_operate(&acc,&self.seg[l].0);if p(&nacc){return Some(self.bisect_perfect(l,acc,p).0);}acc=nacc;l+=1;}l>>=1;k+=1;}for k in(0..k).rev(){let r=r>>k;if r&1!=0{let nacc=M::agg_operate(&acc,&self.seg[r-1].0);if p(&nacc){return Some(self.bisect_perfect(r-1,acc,p).0);}acc=nacc;}}None}#[doc=" Returns the last index that satisfies a accumlative predicate."]pub fn rposition_acc<P>(&mut self,l:usize,r:usize,p:P)->Option<usize>where P:Fn(&M::Agg)->bool{let mut l=l+self.n;let mut r=r+self.n;self.propagate(l,false,true);self.propagate(r,true,true);let mut c=0usize;let mut k=0usize;let mut acc=M::agg_unit();while l>>k<r{c<<=1;if l&1<<k!=0{l+=1<<k;c+=1;}if r&1!=0{r-=1;let nacc=M::agg_operate(&self.seg[r].0,&acc);if p(&nacc){return Some(self.rbisect_perfect(r,acc,p).0);}acc=nacc;}r>>=1;k+=1;}for k in(0..k).rev(){if c&1!=0{l-=1<<k;let l=l>>k;let nacc=M::agg_operate(&self.seg[l].0,&acc);if p(&nacc){return Some(self.rbisect_perfect(l,acc,p).0);}acc=nacc;}c>>=1;}None}}}
pub trait MonoidAction{type Key;type Agg:Clone;type Act:Clone;type AggMonoid:Monoid<T=Self::Agg>;type ActMonoid:Monoid<T=Self::Act>;fn single_agg(key:&Self::Key)->Self::Agg;fn act_key(x:&Self::Key,a:&Self::Act)->Self::Key;fn act_agg(x:&Self::Agg,a:&Self::Act)->Option<Self::Agg>;fn toggle(_x:&mut Self::Agg){}#[inline]fn agg_unit()->Self::Agg{<Self::AggMonoid as Unital>::unit()}#[inline]fn act_unit()->Self::Act{<Self::ActMonoid as Unital>::unit()}#[inline]fn agg_operate(x:&Self::Agg,y:&Self::Agg)->Self::Agg{<Self::AggMonoid as Magma>::operate(x,y)}#[inline]fn act_operate(x:&Self::Act,y:&Self::Act)->Self::Act{<Self::ActMonoid as Magma>::operate(x,y)}#[inline]fn agg_operate_assign(x:&mut Self::Agg,y:&Self::Agg){*x=<Self::AggMonoid as Magma>::operate(x,y);}#[inline]fn act_operate_assign(x:&mut Self::Act,y:&Self::Act){*x=<Self::ActMonoid as Magma>::operate(x,y);}}
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::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::monoid_action_impls::*;
pub mod monoid_action_impls{use super::*;use std::{cmp::Ordering,marker::PhantomData,ops::{Add,Mul,Sub}};pub struct EmptyLazy<M>{_marker:PhantomData<fn()->M>}impl<M>MonoidAction for EmptyLazy<M>where M:Monoid{type Key=M::T;type Agg=M::T;type Act=();type AggMonoid=M;type ActMonoid=();fn single_agg(key:&Self::Key)->Self::Agg{key.clone()}fn act_key(x:&Self::Key,_a:&Self::Act)->Self::Key{x.clone()}fn act_agg(x:&Self::Agg,_a:&Self::Act)->Option<Self::Agg>{Some(x.clone())}}pub struct EmptyAction<T>{_marker:PhantomData<fn()->T>}impl<T>MonoidAction for EmptyAction<T>where T:Clone{type Key=T;type Agg=();type Act=();type AggMonoid=();type ActMonoid=();fn single_agg(_key:&Self::Key)->Self::Agg{}fn act_key(x:&Self::Key,_a:&Self::Act)->Self::Key{x.clone()}fn act_agg(_x:&Self::Agg,_a:&Self::Act)->Option<Self::Agg>{Some(())}}pub struct RangeSumRangeAdd<T>{_marker:PhantomData<fn()->T>}impl<T>MonoidAction for RangeSumRangeAdd<T>where T:Copy+Zero+One+Add<Output=T>+Mul<Output=T>+PartialEq{type Key=T;type Agg=(T,T);type Act=T;type AggMonoid=(AdditiveOperation<T>,AdditiveOperation<T>);type ActMonoid=AdditiveOperation<T>;fn single_agg(key:&Self::Key)->Self::Agg{(*key,T::one())}fn act_key(&x:&Self::Key,&a:&Self::Act)->Self::Key{if<Self::ActMonoid as Unital>::is_unit(&a){x}else{x+a}}fn act_agg(&(x,y):&Self::Agg,&a:&Self::Act)->Option<Self::Agg>{Some(if<Self::ActMonoid as Unital>::is_unit(&a){(x,y)}else{(x+a*y,y)})}}pub struct RangeSumRangeLinear<T>{_marker:PhantomData<fn()->T>}impl<T>MonoidAction for RangeSumRangeLinear<T>where T:Copy+Zero+One+Add<Output=T>+Mul<Output=T>+PartialEq{type Key=T;type Agg=(T,T);type Act=(T,T);type AggMonoid=(AdditiveOperation<T>,AdditiveOperation<T>);type ActMonoid=LinearOperation<T>;fn single_agg(key:&Self::Key)->Self::Agg{(*key,T::one())}fn act_key(&x:&Self::Key,&(a,b):&Self::Act)->Self::Key{if<Self::ActMonoid as Unital>::is_unit(&(a,b)){x}else{a*x+b}}fn act_agg(&(x,y):&Self::Agg,&(a,b):&Self::Act)->Option<Self::Agg>{Some(if<Self::ActMonoid as Unital>::is_unit(&(a,b)){(x,y)}else{(a*x+b*y,y)})}}pub struct RangeSumRangeUpdate<T>{_marker:PhantomData<fn()->T>}impl<T>MonoidAction for RangeSumRangeUpdate<T>where T:Copy+Zero+One+Add<Output=T>+Mul<Output=T>+PartialEq{type Key=T;type Agg=(T,T);type Act=Option<T>;type AggMonoid=(AdditiveOperation<T>,AdditiveOperation<T>);type ActMonoid=LastOperation<T>;fn single_agg(key:&Self::Key)->Self::Agg{(*key,T::one())}fn act_key(&x:&Self::Key,&a:&Self::Act)->Self::Key{a.unwrap_or(x)}fn act_agg(&(x,y):&Self::Agg,a:&Self::Act)->Option<Self::Agg>{Some((a.map(|a|a*y).unwrap_or(x),y))}}pub struct RangeMaxRangeUpdate<T>{_marker:PhantomData<fn()->T>}impl<T>MonoidAction for RangeMaxRangeUpdate<T>where T:Clone+PartialEq+Ord+Bounded{type Key=T;type Agg=T;type Act=Option<T>;type AggMonoid=MaxOperation<T>;type ActMonoid=LastOperation<T>;fn single_agg(key:&Self::Key)->Self::Agg{key.clone()}fn act_key(x:&Self::Key,a:&Self::Act)->Self::Key{a.as_ref().unwrap_or(x).clone()}fn act_agg(x:&Self::Agg,a:&Self::Act)->Option<Self::Agg>{Some(a.as_ref().unwrap_or(x).clone())}}pub struct RangeMinRangeUpdate<T>{_marker:PhantomData<fn()->T>}impl<T>MonoidAction for RangeMinRangeUpdate<T>where T:Clone+PartialEq+Ord+Bounded{type Key=T;type Agg=T;type Act=Option<T>;type AggMonoid=MinOperation<T>;type ActMonoid=LastOperation<T>;fn single_agg(key:&Self::Key)->Self::Agg{key.clone()}fn act_key(x:&Self::Key,a:&Self::Act)->Self::Key{a.as_ref().unwrap_or(x).clone()}fn act_agg(x:&Self::Agg,a:&Self::Act)->Option<Self::Agg>{Some(a.as_ref().unwrap_or(x).clone())}}pub struct RangeMinRangeAdd<T>{_marker:PhantomData<fn()->T>}impl<T>MonoidAction for RangeMinRangeAdd<T>where T:Copy+Ord+Bounded+Zero+Add<Output=T>{type Key=T;type Agg=T;type Act=T;type AggMonoid=MinOperation<T>;type ActMonoid=AdditiveOperation<T>;fn single_agg(key:&Self::Key)->Self::Agg{*key}fn act_key(&x:&Self::Key,&a:&Self::Act)->Self::Key{if<Self::ActMonoid as Unital>::is_unit(&a){x}else{x+a}}fn act_agg(&x:&Self::Agg,&a:&Self::Act)->Option<Self::Agg>{Some(if<Self::ActMonoid as Unital>::is_unit(&a){x}else{x+a})}}#[derive(Debug,Clone,PartialEq,Eq)]pub struct RangeChminChmaxAdd<T>{lb:T,ub:T,bias:T}impl<T>RangeChminChmaxAdd<T>where T:Zero+Bounded{pub fn chmin(x:T)->Self{Self{lb:T::minimum(),ub:x,bias:T::zero()}}pub fn chmax(x:T)->Self{Self{lb:x,ub:T::maximum(),bias:T::zero()}}pub fn add(x:T)->Self{Self{lb:T::minimum(),ub:T::maximum(),bias:x}}}impl<T>Magma for RangeChminChmaxAdd<T>where T:Copy+Zero+One+Ord+Bounded+Add<Output=T>+Sub<Output=T>+Mul<Output=T>+PartialEq{type T=Self;fn operate(x:&Self::T,y:&Self::T)->Self::T{Self{lb:(x.lb+x.bias).min(y.ub).max(y.lb)-x.bias,ub:(x.ub+x.bias).max(y.lb).min(y.ub)-x.bias,bias:x.bias+y.bias}}}impl<T>Associative for RangeChminChmaxAdd<T>where T:Copy+Zero+One+Ord+Bounded+Add<Output=T>+Sub<Output=T>+Mul<Output=T>+PartialEq{}impl<T>Unital for RangeChminChmaxAdd<T>where T:Copy+Zero+One+Ord+Bounded+Add<Output=T>+Sub<Output=T>+Mul<Output=T>+PartialEq{fn unit()->Self::T{Self{lb:T::minimum(),ub:T::maximum(),bias:T::zero()}}}#[derive(Debug,Clone,PartialEq,Eq)]pub struct RangeSumRangeChminChmaxAdd<T>{min:T,max:T,min2:T,max2:T,pub sum:T,size:T,n_min:T,n_max:T}impl<T>RangeSumRangeChminChmaxAdd<T>where T:Copy+Zero+One+Ord+Bounded+Add<Output=T>+Sub<Output=T>+Mul<Output=T>+PartialEq{pub fn single(key:T,size:T)->Self{Self{min:key,max:key,min2:T::maximum(),max2:T::minimum(),sum:key*size,size,n_min:size,n_max:size}}}impl<T>Magma for RangeSumRangeChminChmaxAdd<T>where T:Copy+Zero+One+Ord+Bounded+Add<Output=T>+Sub<Output=T>+Mul<Output=T>+PartialEq{type T=Self;fn operate(x:&Self::T,y:&Self::T)->Self::T{Self{min:x.min.min(y.min),max:x.max.max(y.max),min2:if x.min==y.min{x.min2.min(y.min2)}else if x.min2<=y.min{x.min2}else if y.min2<=x.min{y.min2}else{x.min.max(y.min)},max2:if x.max==y.max{x.max2.max(y.max2)}else if x.max2>=y.max{x.max2}else if y.max2>=x.max{y.max2}else{x.max.min(y.max)},sum:x.sum+y.sum,size:x.size+y.size,n_min:match x.min.cmp(&y.min){Ordering::Less=>x.n_min,Ordering::Equal=>x.n_min+y.n_min,Ordering::Greater=>y.n_min,},n_max:match x.max.cmp(&y.max){Ordering::Less=>y.n_max,Ordering::Equal=>x.n_max+y.n_max,Ordering::Greater=>x.n_max,}}}}impl<T>Associative for RangeSumRangeChminChmaxAdd<T>where T:Copy+Zero+One+Ord+Bounded+Add<Output=T>+Sub<Output=T>+Mul<Output=T>+PartialEq{}impl<T>Unital for RangeSumRangeChminChmaxAdd<T>where T:Copy+Zero+One+Ord+Bounded+Add<Output=T>+Sub<Output=T>+Mul<Output=T>+PartialEq{fn unit()->Self::T{Self{min:T::maximum(),max:T::minimum(),min2:T::maximum(),max2:T::minimum(),sum:T::zero(),size:T::zero(),n_min:T::zero(),n_max:T::zero()}}}impl<T>MonoidAction for RangeSumRangeChminChmaxAdd<T>where T:Copy+Zero+One+Ord+Bounded+Add<Output=T>+Sub<Output=T>+Mul<Output=T>+PartialEq{type Key=T;type Agg=Self;type Act=RangeChminChmaxAdd<T>;type AggMonoid=Self;type ActMonoid=RangeChminChmaxAdd<T>;fn single_agg(&key:&Self::Key)->Self::Agg{Self::single(key,T::one())}fn act_key(&x:&Self::Key,a:&Self::Act)->Self::Key{if<Self::ActMonoid as Unital>::is_unit(a){x}else{x.max(a.lb).min(a.ub)+a.bias}}fn act_agg(x:&Self::Agg,a:&Self::Act)->Option<Self::Agg>{Some(if<Self::ActMonoid as Unital>::is_unit(a){x.clone()}else if x.size.is_zero(){Self::unit()}else if x.min==x.max||a.lb==a.ub||a.lb>=x.max||a.ub<=x.min{Self::single(x.min.max(a.lb).min(a.ub)+a.bias,x.size)}else if x.min2==x.max{let mut x=x.clone();let min=x.min.max(a.lb)+a.bias;let max=x.max.min(a.ub)+a.bias;x.min=min;x.max2=min;x.max=max;x.min2=max;x.sum=min*x.n_min+max*x.n_max;x}else if a.lb<x.min2&&x.max2<a.ub{let mut x=x.clone();let min=x.min.max(a.lb);let max=x.max.min(a.ub);x.sum=x.sum+(min-x.min)*x.n_min+(max-x.max)*x.n_max+a.bias*x.size;x.min=min+a.bias;x.max=max+a.bias;x.min2=x.min2+a.bias;x.max2=x.max2+a.bias;x}else{return None;})}}}
pub use self::last_operation_impl::LastOperation;
mod last_operation_impl{use super::*;use std::marker::PhantomData;#[doc=" retain the last element"]pub struct LastOperation<T:Clone>{_marker:PhantomData<fn()->T>}impl<T:Clone>Magma for LastOperation<T>{type T=Option<T>;#[inline]fn operate(x:&Self::T,y:&Self::T)->Self::T{y.as_ref().or(x.as_ref()).cloned()}}impl<T:Clone>Unital for LastOperation<T>{#[inline]fn unit()->Self::T{None}}impl<T:Clone>Associative for LastOperation<T>{}impl<T:Clone>Idempotent for LastOperation<T>{}}
pub use self::linear_operation_impl::LinearOperation;
mod linear_operation_impl{use super::*;use std::{marker::PhantomData,ops::{Add,Div,Mul,Neg,Sub}};#[doc=" $(a, b) \\circ (c, d) = \\lambda x. c \\times (a \\times x + b) + d$"]pub struct LinearOperation<T:Clone+Zero+Add<Output=T>+One+Mul<Output=T>>{_marker:PhantomData<fn()->T>}impl<T:Clone+Zero+One+Add<Output=T>+Mul<Output=T>>Magma for LinearOperation<T>{type T=(T,T);#[inline]fn operate(x:&Self::T,y:&Self::T)->Self::T{(y.0.clone()*x.0.clone(),y.0.clone()*x.1.clone()+y.1.clone())}}impl<T:Clone+Zero+One+Add<Output=T>+Mul<Output=T>>Unital for LinearOperation<T>{#[inline]fn unit()->Self::T{(One::one(),Zero::zero())}}impl<T:Clone+Zero+One+Add<Output=T>+Mul<Output=T>>Associative for LinearOperation<T>{}impl<T:Clone+Zero+One+Add<Output=T>+Sub<Output=T>+Neg<Output=T>+Mul<Output=T>+Div<Output=T>>Invertible for LinearOperation<T>{fn inverse(x:&Self::T)->Self::T{let y=<T as One>::one().div(x.0.clone());(y.clone(),-y.mul(x.1.clone()))}}}
pub use self::max_operation_impl::MaxOperation;
mod max_operation_impl{use super::*;use std::marker::PhantomData;#[doc=" binary operation to select larger element"]pub struct MaxOperation<T:Clone+Ord+Bounded>{_marker:PhantomData<fn()->T>}impl<T:Clone+Ord+Bounded>Magma for MaxOperation<T>{type T=T;#[inline]fn operate(x:&Self::T,y:&Self::T)->Self::T{x.max(y).clone()}}impl<T:Clone+Ord+Bounded>Unital for MaxOperation<T>{#[inline]fn unit()->Self::T{<T as Bounded>::minimum()}}impl<T:Clone+Ord+Bounded>Associative for MaxOperation<T>{}impl<T:Clone+Ord+Bounded>Commutative for MaxOperation<T>{}impl<T:Clone+Ord+Bounded>Idempotent for MaxOperation<T>{}}
pub use self::min_operation_impl::MinOperation;
mod min_operation_impl{use super::*;use std::marker::PhantomData;#[doc=" binary operation to select smaller element"]pub struct MinOperation<T:Clone+Ord+Bounded>{_marker:PhantomData<fn()->T>}impl<T:Clone+Ord+Bounded>Magma for MinOperation<T>{type T=T;#[inline]fn operate(x:&Self::T,y:&Self::T)->Self::T{x.min(y).clone()}}impl<T:Clone+Ord+Bounded>Unital for MinOperation<T>{#[inline]fn unit()->Self::T{<T as Bounded>::maximum()}}impl<T:Clone+Ord+Bounded>Associative for MinOperation<T>{}impl<T:Clone+Ord+Bounded>Commutative for MinOperation<T>{}impl<T:Clone+Ord+Bounded>Idempotent for MinOperation<T>{}}
mod tuple_operation_impl{#![allow(unused_variables,clippy::unused_unit)]use super::*;macro_rules!impl_tuple_operation{(@impl$($T:ident)*,$($i:tt)*)=>{impl<$($T:Magma),*>Magma for($($T,)*){type T=($(<$T as Magma>::T,)*);#[inline]fn operate(x:&Self::T,y:&Self::T)->Self::T{($(<$T as Magma>::operate(&x.$i,&y.$i),)*)}}impl<$($T:Unital),*>Unital for($($T,)*){#[inline]fn unit()->Self::T{($(<$T as Unital>::unit(),)*)}}impl<$($T:Associative),*>Associative for($($T,)*){}impl<$($T:Commutative),*>Commutative for($($T,)*){}impl<$($T:Idempotent),*>Idempotent for($($T,)*){}impl<$($T:Invertible),*>Invertible for($($T,)*){#[inline]fn inverse(x:&Self::T)->Self::T{($(<$T as Invertible>::inverse(&x.$i),)*)}}};(@inner[$($T:ident)*][][$($i:tt)*][])=>{impl_tuple_operation!(@impl$($T)*,$($i)*);};(@inner[$($T:ident)*][$U:ident$($Rest:ident)*][$($i:tt)*][$j:tt$($rest:tt)*])=>{impl_tuple_operation!(@impl$($T)*,$($i)*);impl_tuple_operation!(@inner[$($T)*$U][$($Rest)*][$($i)*$j][$($rest)*]);};($($T:ident)*,$($i:tt)*)=>{impl_tuple_operation!(@inner[][$($T)*][][$($i)*]);};}impl_tuple_operation!(A B C D E F G H I J,0 1 2 3 4 5 6 7 8 9);}
pub use self::bounded::Bounded;
mod bounded{#[doc=" Trait for max/min bounds"]pub trait Bounded:Sized+PartialOrd{fn maximum()->Self;fn minimum()->Self;fn is_maximum(&self)->bool{self==&Self::maximum()}fn is_minimum(&self)->bool{self==&Self::minimum()}fn set_maximum(&mut self){*self=Self::maximum()}fn set_minimum(&mut self){*self=Self::minimum()}}macro_rules!bounded_num_impls{($($t:ident)*)=>{$(impl Bounded for$t{fn maximum()->Self{std::$t::MAX}fn minimum()->Self{std::$t::MIN}})*};}bounded_num_impls!(u8 u16 u32 u64 u128 usize i8 i16 i32 i64 i128 isize f32 f64);macro_rules!bounded_tuple_impls{(@impl$($T:ident)*)=>{impl<$($T:Bounded),*>Bounded for($($T,)*){fn maximum()->Self{($(<$T as Bounded>::maximum(),)*)}fn minimum()->Self{($(<$T as Bounded>::minimum(),)*)}}};(@inner$($T:ident)*,)=>{bounded_tuple_impls!(@impl$($T)*);};(@inner$($T:ident)*,$U:ident$($Rest:ident)*)=>{bounded_tuple_impls!(@impl$($T)*);bounded_tuple_impls!(@inner$($T)*$U,$($Rest)*);};($T:ident$($Rest:ident)*)=>{bounded_tuple_impls!(@inner$T,$($Rest)*);};}bounded_tuple_impls!(A B C D E F G H I J);impl Bounded for(){fn maximum()->Self{}fn minimum()->Self{}}impl Bounded for bool{fn maximum()->Self{true}fn minimum()->Self{false}}impl<T>Bounded for Option<T>where T:Bounded{fn maximum()->Self{Some(<T as Bounded>::maximum())}fn minimum()->Self{None}}impl<T>Bounded for std::cmp::Reverse<T>where T:Bounded{fn maximum()->Self{std::cmp::Reverse(<T as Bounded>::minimum())}fn minimum()->Self{std::cmp::Reverse(<T as Bounded>::maximum())}}}