pub fn main() {
    crate::prepare!();
    sc!(n, m, k);
    let mut esper = EsperEstimator::new(
        |&(_n, m, k): &(usize, usize, usize)| m.cmp(&k),
        |&(n, m, k): &(usize, usize, usize)| {
            let f = vec![
                M::one(),
                M::from(n),
                M::from(m),
                M::from(k),
                M::from(m - 1).pow(n * 2),
                M::from(m).pow(n * 2),
                M::from(m + 1).pow(n * 2),
                M::from(k - 1).pow(n * 2),
                M::from(k).pow(n * 2),
                M::from(k + 1).pow(n * 2),
            ];
            let mut nf = vec![];
            for i in 0..f.len() {
                for j in i + 1..f.len() {
                    nf.push(f[i] * f[j]);
                }
            }
            nf
        },
    );
    for m in 1..=10 {
        let mut dp = vec![vec![M::zero(); m]; m];
        for i in 0..m {
            dp[i][i] = M::one();
        }
        for n in 2..=10 {
            let mut ndp = vec![vec![M::zero(); m]; m];
            for i in 0..m {
                for j in 0..m {
                    for v in 0..m {
                        ndp[i.min(v)][j.max(v)] += dp[i][j];
                    }
                }
            }
            dp = ndp;
            if n % 2 == 0 {
                for k in 1..m + 2 {
                    let mut res = M::zero();
                    for i in 0..m {
                        for j in i..m {
                            if j - i < k {
                                res += dp[i][j];
                            }
                        }
                    }
                    esper.push((n / 2, m, k), res);
                }
            }
        }
    }
    let esper = esper.solve_checked();
    pp!(esper.solve((n, m, k)));
}
pub type M = mint_basic::MInt998244353;
#[derive(Debug, Clone)]
pub struct EsperEstimator<T, Input, Class, FC, FF>
where
    Class: Eq + Hash,
    FC: Fn(&Input) -> Class,
    FF: Fn(&Input) -> Vec<T>,
{
    class: FC,
    feature: FF,
    data: HashMap<Class, (Vec<Vec<T>>, Vec<T>)>,
    _marker: PhantomData<fn() -> (T, Input, Class)>,
}
#[derive(Debug, Clone)]
pub struct EsperSolver<T, Input, Class, FC, FF>
where
    Class: Eq + Hash,
    FC: Fn(&Input) -> Class,
    FF: Fn(&Input) -> Vec<T>,
{
    class: FC,
    feature: FF,
    data: HashMap<Class, Option<Vec<T>>>,
    _marker: PhantomData<fn() -> (T, Input, Class)>,
}
impl<T, Input, Class, FC, FF> EsperEstimator<T, Input, Class, FC, FF>
where
    Class: Eq + Hash,
    FC: Fn(&Input) -> Class,
    FF: Fn(&Input) -> Vec<T>,
{
    pub fn new(class: FC, feature: FF) -> Self {
        Self {
            class,
            feature,
            data: Default::default(),
            _marker: PhantomData,
        }
    }
    pub fn push(&mut self, input: Input, output: T) {
        let class = (self.class)(&input);
        let feature = (self.feature)(&input);
        let entry = self.data.entry(class).or_default();
        entry.0.push(feature);
        entry.1.push(output);
    }
}
impl<T, Input, Class, FC, FF> EsperEstimator<T, Input, Class, FC, FF>
where
    Class: Eq + Hash,
    T: Copy + PartialEq + Zero + One + Sub<Output = T> + Mul<Output = T> + Div<Output = T>,
    FC: Fn(&Input) -> Class,
    FF: Fn(&Input) -> Vec<T>,
{
    pub fn solve(self) -> EsperSolver<T, Input, Class, FC, FF> {
        let data: HashMap<_, _> = self
            .data
            .into_iter()
            .map(|(key, (a, b))| {
                (
                    key,
                    Matrix::from_vec(a).solve_system_of_linear_equations(&b),
                )
            })
            .collect();
        EsperSolver {
            class: self.class,
            feature: self.feature,
            data,
            _marker: PhantomData,
        }
    }
    pub fn solve_checked(self) -> EsperSolver<T, Input, Class, FC, FF>
    where
        Class: Debug,
        T: Debug,
    {
        let data: HashMap<_, _> = self
            .data
            .into_iter()
            .map(|(key, (a, b))| {
                let mat = Matrix::from_vec(a.clone());
                let coeff = mat.solve_system_of_linear_equations(&b);
                if coeff.is_none() {
                    eprintln!(
                        "failed to solve linear equations: key={:?} A={:?} b={:?}",
                        key, &mat.data, &b
                    );
                }
                (key, coeff)
            })
            .collect();
        EsperSolver {
            class: self.class,
            feature: self.feature,
            data,
            _marker: PhantomData,
        }
    }
}
impl<T, Input, Class, FC, FF> EsperSolver<T, Input, Class, FC, FF>
where
    Class: Eq + Hash,
    T: Copy + Zero + Add<Output = T> + Mul<Output = T>,
    FC: Fn(&Input) -> Class,
    FF: Fn(&Input) -> Vec<T>,
{
    pub fn solve(&self, input: Input) -> T {
        let coeff = self
            .data
            .get(&(self.class)(&input))
            .expect("unrecognized class")
            .as_ref()
            .expect("failed to solve");
        let feature = (self.feature)(&input);
        feature
            .into_iter()
            .zip(coeff)
            .map(|(x, &y)| x * y)
            .fold(T::zero(), |x, y| x + y)
    }
}
use std::{
    fmt::Debug,
    hash::Hash,
    marker::PhantomData,
    ops::{Add, Div, Mul, Sub},
};
#[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 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 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 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::matrix::Matrix;
mod matrix{use super::{One,Zero};use std::ops::{Add,Div,Index,IndexMut,Mul,Sub};#[derive(Clone,Debug,PartialEq,Eq)]pub struct Matrix<T>{pub shape:(usize,usize),pub data:Vec<Vec<T>>}impl<T:Clone>Matrix<T>{pub fn new(shape:(usize,usize),z:T)->Self{Self{shape,data:vec![vec![z;shape.1];shape.0]}}}impl<T>Matrix<T>{pub fn from_vec(data:Vec<Vec<T>>)->Self{Self{shape:(data.len(),data.first().map(Vec::len).unwrap_or_default()),data}}}impl<T>Matrix<T>where T:Clone+Zero{pub fn zeros(shape:(usize,usize))->Self{Self{shape,data:vec![vec![Zero::zero();shape.1];shape.0]}}}impl<T>Matrix<T>where T:Clone+Zero+One{pub fn eye(shape:(usize,usize))->Self{let mut data=vec![vec![Zero::zero();shape.1];shape.0];for(i,d)in data.iter_mut().enumerate(){d[i]=One::one();}Self{shape,data}}}impl<T>Index<usize>for Matrix<T>{type Output=Vec<T>;fn index(&self,index:usize)->&Self::Output{&self.data[index]}}impl<T>IndexMut<usize>for Matrix<T>{fn index_mut(&mut self,index:usize)->&mut Self::Output{&mut self.data[index]}}impl<T>Index<(usize,usize)>for Matrix<T>{type Output=T;fn index(&self,index:(usize,usize))->&Self::Output{&self.data[index.0][index.1]}}impl<T>IndexMut<(usize,usize)>for Matrix<T>{fn index_mut(&mut self,index:(usize,usize))->&mut Self::Output{&mut self.data[index.0][index.1]}}impl<'a,T>Add for&'a Matrix<T>where T:Copy+Zero+Add<Output=T>{type Output=Matrix<T>;fn add(self,rhs:Self)->Self::Output{assert_eq!(self.shape,rhs.shape);let mut res=Matrix::zeros(self.shape);for i in 0..self.shape.0{for j in 0..self.shape.1{res[i][j]=self[i][j]+rhs[i][j];}}res}}impl<'a,T>Sub for&'a Matrix<T>where T:Copy+Zero+Sub<Output=T>{type Output=Matrix<T>;fn sub(self,rhs:Self)->Self::Output{assert_eq!(self.shape,rhs.shape);let mut res=Matrix::zeros(self.shape);for i in 0..self.shape.0{for j in 0..self.shape.1{res[i][j]=self[i][j]-rhs[i][j];}}res}}impl<'a,T>Mul for&'a Matrix<T>where T:Copy+Zero+Add<Output=T>+Mul<Output=T>{type Output=Matrix<T>;fn mul(self,rhs:Self)->Self::Output{assert_eq!(self.shape.1,rhs.shape.0);let mut res=Matrix::zeros((self.shape.0,rhs.shape.1));for i in 0..self.shape.0{for j in 0..rhs.shape.1{for k in 0..self.shape.1{res[i][j]=res[i][j]+self[i][k]*rhs[k][j];}}}res}}impl<T>Matrix<T>where T:Copy+Zero+One+Add<Output=T>+Mul<Output=T>{pub fn pow(&self,mut n:usize)->Self{assert_eq!(self.shape.0,self.shape.1);let mut x=self.clone();let mut res=Matrix::eye(self.shape);while n>0{if n&1==1{res=&res*&x;}x=&x*&x;n>>=1;}res}}impl<T>Matrix<T>where T:Copy+PartialEq+Zero+One+Sub<Output=T>+Mul<Output=T>+Div<Output=T>{pub fn row_reduction(&mut self,normalize:bool){let(n,m)=self.shape;let mut c=0;for r in 0..n{loop{if c>=m{return;}if let Some(pivot)=(r..n).find(|&p|!self[p][c].is_zero()){self.data.swap(r,pivot);break;};c+=1;}let d=T::one()/self[r][c];if normalize{for j in c..m{self[r][j]=self[r][j]*d;}}for i in(0..n).filter(|&i|i!=r){let mut e=self[i][c];if!normalize{e=e*d;}for j in c..m{self[i][j]=self[i][j]-e*self[r][j];}}c+=1;}}pub fn rank(&mut self)->usize{let n=self.shape.0;self.row_reduction(false);(0..n).filter(|&i|!self.data[i].iter().all(|x|x.is_zero())).count()}pub fn determinant(&mut self)->T{assert_eq!(self.shape.0,self.shape.1);self.row_reduction(false);let mut d=T::one();for i in 0..self.shape.0{d=d*self[i][i];}d}pub fn solve_system_of_linear_equations(&self,b:&[T])->Option<Vec<T>>{assert_eq!(self.shape.0,b.len());let(n,m)=self.shape;let mut c=Matrix::<T>::zeros((n,m+1));for i in 0..n{c[i][..m].clone_from_slice(&self[i]);c[i][m]=b[i];}c.row_reduction(true);let mut x=vec![T::zero();m];for i in 0..n{let mut j=0usize;while j<=m&&c[i][j].is_zero(){j+=1;}if j==m{return None;}if j<m{x[j]=c[i][m];}}Some(x)}pub fn inverse(&self)->Option<Matrix<T>>{assert_eq!(self.shape.0,self.shape.1);let n=self.shape.0;let mut c=Matrix::<T>::zeros((n,n*2));for i in 0..n{c[i][..n].clone_from_slice(&self[i]);c[i][n+i]=T::one();}c.row_reduction(true);if(0..n).any(|i|c[i][i].is_zero()){None}else{Some(Self::from_vec(c.data.into_iter().map(|r|r[n..].to_vec()).collect()))}}}}