fn main() {
    #![allow(unused_imports, unused_macros)]
    prepare_io!(_in_buf, scanner, _out);
    macro_rules ! print { ($ ($ arg : tt) *) => (:: std :: write ! (_out , $ ($ arg) *) . expect ("io error")) }
    macro_rules ! println { ($ ($ arg : tt) *) => (:: std :: writeln ! (_out , $ ($ arg) *) . expect ("io error")) }
    scan!(scanner, n, (g, _): {TreeGraphScanner::<Usize1>::new(n)});
    let r =
        ReRooting::<(AdditiveOperation<usize>, AdditiveOperation<usize>), _>::new(&g, |d, _, o| {
            (d.0 + d.1, d.1 + o.is_some() as usize)
        });
    let distsum = (0..n).map(|u| r.dp[u].0).sum::<usize>();
    let ans = M::from(distsum) / (M::from(n - 1) * M::from(n) * M::from(n - 1));
    println!("{}", M::one() - ans);
}
pub type M = mint_basic::MInt998244353;
#[macro_export]
macro_rules! prepare_io {
    ($ in_buf : ident , $ scanner : ident , $ out : ident) => {
        use std::io::{stdout, BufWriter, Write as _};
        let $in_buf = read_stdin_all_unchecked();
        let mut $scanner = Scanner::new(&$in_buf);
        let $out = stdout();
        let mut $out = BufWriter::new($out.lock());
    };
}
pub fn echo<T: std::fmt::Display>(
    mut writer: impl std::io::Write,
    iter: impl IntoIterator<Item = T>,
    sep: impl std::fmt::Display,
) -> std::io::Result<()> {
    let mut iter = iter.into_iter();
    if let Some(item) = iter.next() {
        write!(writer, "{}", item)?;
    }
    for item in iter {
        write!(writer, "{}{}", sep, item)?;
    }
    writeln!(writer)
}
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_stdin_line() -> String {
    let mut s = String::new();
    std::io::stdin().read_line(&mut s).expect("io error");
    s
}
pub use scanner_impls::{IterScan, MarkedIterScan, Scanner};
mod scanner_impls {
    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 { ($ ($ 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) ?,) *)) } } } ; }
    iter_scan_tuple_impl!();
    iter_scan_tuple_impl!(A);
    iter_scan_tuple_impl ! (A B);
    iter_scan_tuple_impl ! (A B C);
    iter_scan_tuple_impl ! (A B C D);
    iter_scan_tuple_impl ! (A B C D E);
    iter_scan_tuple_impl ! (A B C D E F);
    iter_scan_tuple_impl ! (A B C D E F G);
    iter_scan_tuple_impl ! (A B C D E F G H);
    iter_scan_tuple_impl ! (A B C D E F G H I);
    iter_scan_tuple_impl ! (A B C D E F G H I J);
    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)
        }
    }
}
pub use marker_impls::{CharWithBase, Chars, CharsWithBase, Collect, SizedCollect, Usize1};
mod marker_impls {
    use super::*;
    use std::{
        iter::{repeat_with, FromIterator},
        marker::PhantomData,
    };
    #[derive(Debug, Copy, Clone)]
    pub struct Usize1;
    impl IterScan for Usize1 {
        type Output = usize;
        #[inline]
        fn scan<'a, I: Iterator<Item = &'a str>>(iter: &mut I) -> Option<Self::Output> {
            <usize as IterScan>::scan(iter)?.checked_sub(1)
        }
    }
    #[derive(Debug, Copy, Clone)]
    pub struct CharWithBase(pub char);
    impl MarkedIterScan for CharWithBase {
        type Output = usize;
        #[inline]
        fn mscan<'a, I: Iterator<Item = &'a str>>(self, iter: &mut I) -> Option<Self::Output> {
            Some((<char as IterScan>::scan(iter)? as u8 - self.0 as u8) as usize)
        }
    }
    #[derive(Debug, Copy, Clone)]
    pub struct Chars;
    impl IterScan for Chars {
        type Output = Vec<char>;
        #[inline]
        fn scan<'a, I: Iterator<Item = &'a str>>(iter: &mut I) -> Option<Self::Output> {
            Some(iter.next()?.chars().collect())
        }
    }
    #[derive(Debug, Copy, Clone)]
    pub struct CharsWithBase(pub char);
    impl MarkedIterScan for CharsWithBase {
        type Output = Vec<usize>;
        #[inline]
        fn mscan<'a, I: Iterator<Item = &'a str>>(self, iter: &mut I) -> Option<Self::Output> {
            Some(
                iter.next()?
                    .chars()
                    .map(|c| (c as u8 - self.0 as u8) as usize)
                    .collect(),
            )
        }
    }
    #[derive(Debug, Copy, Clone)]
    pub struct 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()
        }
    }
}
#[macro_export]
macro_rules ! scan_value { ($ scanner : expr , ($ ($ t : tt) ,*)) => { ($ ($ crate :: scan_value ! ($ scanner , $ t)) ,*) } ; ($ scanner : expr , [$ t : tt ; $ len : expr]) => { (0 ..$ len) . map (| _ | $ crate :: scan_value ! ($ scanner , $ t)) . collect ::< Vec < _ >> () } ; ($ scanner : expr , [$ t : ty ; $ len : expr]) => { $ scanner . scan_vec ::<$ t > ($ len) } ; ($ scanner : expr , [$ t : ty]) => { $ scanner . iter ::<$ t > () } ; ($ scanner : expr , { $ e : expr }) => { $ scanner . mscan ($ e) } ; ($ scanner : expr , $ t : ty) => { $ scanner . scan ::<$ t > () } ; }
#[macro_export]
macro_rules ! scan { ($ scanner : expr) => { } ; ($ scanner : expr ,) => { } ; ($ scanner : expr , mut $ var : tt : $ t : tt) => { let mut $ var = $ crate :: scan_value ! ($ scanner , $ t) ; } ; ($ scanner : expr , $ var : tt : $ t : tt) => { let $ var = $ crate :: scan_value ! ($ scanner , $ t) ; } ; ($ scanner : expr , mut $ var : tt : $ t : tt , $ ($ rest : tt) *) => { let mut $ var = $ crate :: scan_value ! ($ scanner , $ t) ; scan ! ($ scanner , $ ($ rest) *) } ; ($ scanner : expr , $ var : tt : $ t : tt , $ ($ rest : tt) *) => { let $ var = $ crate :: scan_value ! ($ scanner , $ t) ; scan ! ($ scanner , $ ($ rest) *) } ; ($ scanner : expr , mut $ var : tt) => { let mut $ var = $ crate :: scan_value ! ($ scanner , usize) ; } ; ($ scanner : expr , $ var : tt) => { let $ var = $ crate :: scan_value ! ($ scanner , usize) ; } ; ($ scanner : expr , mut $ var : tt , $ ($ rest : tt) *) => { let mut $ var = $ crate :: scan_value ! ($ scanner , usize) ; scan ! ($ scanner , $ ($ rest) *) } ; ($ scanner : expr , $ var : tt , $ ($ rest : tt) *) => { let $ var = $ crate :: scan_value ! ($ scanner , usize) ; scan ! ($ scanner , $ ($ rest) *) } ; }
pub use sparse_graph::{
    Adjacency, BidirectionalGraphScanner, BidirectionalSparseGraph, DirectedGraphScanner,
    DirectedSparseGraph, SparseGraph, TreeGraphScanner, UndirectedGraphScanner,
    UndirectedSparseGraph,
};
pub mod sparse_graph {
    use super::*;
    use std::{iter, marker::PhantomData, ops, slice};
    type Marker<T> = PhantomData<fn() -> T>;
    #[derive(Clone, Copy, Debug, Default, Eq, PartialEq, Ord, PartialOrd, Hash)]
    pub struct DirectedEdge;
    #[derive(Clone, Copy, Debug, Default, Eq, PartialEq, Ord, PartialOrd, Hash)]
    pub struct UndirectedEdge;
    #[derive(Clone, Copy, Debug, Default, Eq, PartialEq, Ord, PartialOrd, Hash)]
    pub struct BidirectionalEdge;
    #[derive(Clone, Copy, Debug, Default, Eq, PartialEq, Ord, PartialOrd, Hash)]
    pub struct Adjacency {
        pub id: usize,
        pub to: usize,
    }
    impl Adjacency {
        pub fn new(id: usize, to: usize) -> Adjacency {
            Adjacency { id, to }
        }
    }
    /// Static Sparse Graph represented as Compressed Sparse Row.
    #[derive(Debug, Clone)]
    pub struct SparseGraph<D> {
        vsize: usize,
        pub start: Vec<usize>,
        pub elist: Vec<Adjacency>,
        pub edges: Vec<(usize, usize)>,
        _marker: Marker<D>,
    }
    impl<D> SparseGraph<D> {
        /// Return the number of vertices.
        pub fn vertices_size(&self) -> usize {
            self.vsize
        }
        /// Return the number of edges.
        pub fn edges_size(&self) -> usize {
            self.edges.len()
        }
        /// Return an iterator over graph vertices.
        pub fn vertices(&self) -> ops::Range<usize> {
            0..self.vertices_size()
        }
        /// Return a slice of adjacency vertices.
        pub fn adjacencies(&self, v: usize) -> slice::Iter<'_, Adjacency> {
            self.elist[self.start[v]..self.start[v + 1]].iter()
        }
    }
    pub trait SparseGraphConstruction: Sized {
        fn construct_graph(vsize: usize, edges: Vec<(usize, usize)>) -> SparseGraph<Self>;
    }
    impl<D: SparseGraphConstruction> SparseGraph<D> {
        /// Construct graph from edges.
        pub fn from_edges(vsize: usize, edges: Vec<(usize, usize)>) -> Self {
            D::construct_graph(vsize, edges)
        }
    }
    impl SparseGraphConstruction for DirectedEdge {
        fn construct_graph(vsize: usize, edges: Vec<(usize, usize)>) -> SparseGraph<Self> {
            let mut start: Vec<_> = iter::repeat(0).take(vsize + 1).collect();
            let mut elist = Vec::with_capacity(edges.len());
            unsafe { elist.set_len(edges.len()) }
            for (from, _) in edges.iter().cloned() {
                start[from] += 1;
            }
            for i in 1..=vsize {
                start[i] += start[i - 1];
            }
            for (id, (from, to)) in edges.iter().cloned().enumerate() {
                start[from] -= 1;
                elist[start[from]] = Adjacency::new(id, to);
            }
            SparseGraph {
                vsize,
                start,
                elist,
                edges,
                _marker: PhantomData,
            }
        }
    }
    impl SparseGraphConstruction for UndirectedEdge {
        fn construct_graph(vsize: usize, edges: Vec<(usize, usize)>) -> SparseGraph<Self> {
            let mut start: Vec<_> = iter::repeat(0).take(vsize + 1).collect();
            let mut elist = Vec::with_capacity(edges.len() * 2);
            unsafe { elist.set_len(edges.len() * 2) }
            for (from, to) in edges.iter().cloned() {
                start[to] += 1;
                start[from] += 1;
            }
            for i in 1..=vsize {
                start[i] += start[i - 1];
            }
            for (id, (from, to)) in edges.iter().cloned().enumerate() {
                start[from] -= 1;
                elist[start[from]] = Adjacency::new(id, to);
                start[to] -= 1;
                elist[start[to]] = Adjacency::new(id, from);
            }
            SparseGraph {
                vsize,
                start,
                elist,
                edges,
                _marker: PhantomData,
            }
        }
    }
    impl SparseGraphConstruction for BidirectionalEdge {
        fn construct_graph(vsize: usize, edges: Vec<(usize, usize)>) -> SparseGraph<Self> {
            let mut start: Vec<_> = iter::repeat(0).take(vsize + 1).collect();
            let mut elist = Vec::with_capacity(edges.len() * 2);
            unsafe { elist.set_len(edges.len() * 2) }
            for (from, to) in edges.iter().cloned() {
                start[to] += 1;
                start[from] += 1;
            }
            for i in 1..=vsize {
                start[i] += start[i - 1];
            }
            for (id, (from, to)) in edges.iter().cloned().enumerate() {
                start[from] -= 1;
                elist[start[from]] = Adjacency::new(id * 2, to);
                start[to] -= 1;
                elist[start[to]] = Adjacency::new(id * 2 + 1, from);
            }
            SparseGraph {
                vsize,
                start,
                elist,
                edges,
                _marker: PhantomData,
            }
        }
    }
    pub type DirectedSparseGraph = SparseGraph<DirectedEdge>;
    pub type UndirectedSparseGraph = SparseGraph<UndirectedEdge>;
    pub type BidirectionalSparseGraph = SparseGraph<BidirectionalEdge>;
    pub struct SparseGraphScanner<U: IterScan<Output = usize>, T: IterScan, D> {
        vsize: usize,
        esize: usize,
        _marker: Marker<(U, T, D)>,
    }
    impl<U: IterScan<Output = usize>, T: IterScan, D> SparseGraphScanner<U, T, D> {
        pub fn new(vsize: usize, esize: usize) -> Self {
            Self {
                vsize,
                esize,
                _marker: PhantomData,
            }
        }
    }
    impl<U: IterScan<Output = usize>, T: IterScan, D: SparseGraphConstruction> MarkedIterScan
        for SparseGraphScanner<U, T, D>
    {
        type Output = (SparseGraph<D>, Vec<<T as IterScan>::Output>);
        fn mscan<'a, I: Iterator<Item = &'a str>>(self, iter: &mut I) -> Option<Self::Output> {
            let mut edges = Vec::with_capacity(self.esize);
            let mut rest = Vec::with_capacity(self.esize);
            for _ in 0..self.esize {
                edges.push((U::scan(iter)?, U::scan(iter)?));
                rest.push(T::scan(iter)?);
            }
            let graph = SparseGraph::from_edges(self.vsize, edges);
            Some((graph, rest))
        }
    }
    pub type DirectedGraphScanner<U, T = ()> = SparseGraphScanner<U, T, DirectedEdge>;
    pub type UndirectedGraphScanner<U, T = ()> = SparseGraphScanner<U, T, UndirectedEdge>;
    pub type BidirectionalGraphScanner<U, T = ()> = SparseGraphScanner<U, T, BidirectionalEdge>;
    pub struct TreeGraphScanner<U: IterScan<Output = usize>, T: IterScan = ()> {
        vsize: usize,
        _marker: Marker<(U, T)>,
    }
    impl<U: IterScan<Output = usize>, T: IterScan> TreeGraphScanner<U, T> {
        pub fn new(vsize: usize) -> Self {
            Self {
                vsize,
                _marker: PhantomData,
            }
        }
    }
    impl<U: IterScan<Output = usize>, T: IterScan> MarkedIterScan for TreeGraphScanner<U, T> {
        type Output = (UndirectedSparseGraph, Vec<<T as IterScan>::Output>);
        fn mscan<'a, I: Iterator<Item = &'a str>>(self, iter: &mut I) -> Option<Self::Output> {
            UndirectedGraphScanner::<U, T>::new(self.vsize, self.vsize - 1).mscan(iter)
        }
    }
}
/// dynamic programming on all-rooted trees
///
/// caluculate all subtrees (hanging on the edge) in specific ordering,
/// each subtree calculated in the order of merge and rooting
#[derive(Clone, Debug)]
pub struct ReRooting<'a, M: Monoid, F: Fn(&M::T, usize, Option<usize>) -> M::T> {
    graph: &'a UndirectedSparseGraph,
    /// dp\[v\]: result of v-rooted tree
    pub dp: Vec<M::T>,
    /// ep\[e\]: result of e-subtree, if e >= n then reversed-e-subtree
    pub ep: Vec<M::T>,
    /// rooting(data, vid, (Optional)eid): add root node(vid), result subtree is edge(eid)
    rooting: F,
}
impl<'a, M, F> ReRooting<'a, M, F>
where
    M: Monoid,
    F: Fn(&M::T, usize, Option<usize>) -> M::T,
{
    pub fn new(graph: &'a UndirectedSparseGraph, rooting: F) -> Self {
        let dp = vec![M::unit(); graph.vertices_size()];
        let ep = vec![M::unit(); graph.vertices_size() * 2];
        let mut self_ = Self {
            graph,
            dp,
            ep,
            rooting,
        };
        self_.rerooting();
        self_
    }
    #[inline]
    fn eidx(&self, u: usize, a: &Adjacency) -> usize {
        a.id + self.graph.edges_size() * (u > a.to) as usize
    }
    #[inline]
    fn reidx(&self, u: usize, a: &Adjacency) -> usize {
        a.id + self.graph.edges_size() * (u < a.to) as usize
    }
    #[inline]
    fn merge(&self, x: &M::T, y: &M::T) -> M::T {
        M::operate(x, y)
    }
    #[inline]
    fn add_subroot(&self, x: &M::T, vid: usize, eid: usize) -> M::T {
        (self.rooting)(x, vid, Some(eid))
    }
    #[inline]
    fn add_root(&self, x: &M::T, vid: usize) -> M::T {
        (self.rooting)(x, vid, None)
    }
    fn dfs(&mut self, pa: &Adjacency, p: usize) {
        let u = pa.to;
        let pi = self.eidx(p, pa);
        for a in self.graph.adjacencies(u).filter(|a| a.to != p) {
            let i = self.eidx(u, a);
            self.dfs(a, u);
            self.ep[pi] = self.merge(&self.ep[pi], &self.ep[i]);
        }
        self.ep[pi] = self.add_subroot(&self.ep[pi], u, pa.id);
    }
    fn efs(&mut self, u: usize, p: usize) {
        let m = self.graph.adjacencies(u).len();
        let mut left = vec![M::unit(); m + 1];
        let mut right = vec![M::unit(); m + 1];
        for (k, a) in self.graph.adjacencies(u).enumerate() {
            let i = self.eidx(u, a);
            left[k + 1] = self.merge(&left[k], &self.ep[i]);
        }
        for (k, a) in self.graph.adjacencies(u).enumerate().rev() {
            let i = self.eidx(u, a);
            right[k] = self.merge(&right[k + 1], &self.ep[i]);
        }
        self.dp[u] = self.add_root(&left[m], u);
        for (k, a) in self.graph.adjacencies(u).enumerate() {
            if a.to != p {
                let i = self.reidx(u, a);
                self.ep[i] = self.merge(&left[k], &right[k + 1]);
                self.ep[i] = self.add_subroot(&self.ep[i], u, a.id);
                self.efs(a.to, u);
            }
        }
    }
    fn rerooting(&mut self) {
        for a in self.graph.adjacencies(0) {
            self.dfs(a, 0);
        }
        self.efs(0, std::usize::MAX);
    }
}
/// binary operaion: $T \circ T \to T$
pub trait Magma {
    /// type of operands: $T$
    type T: Clone;
    /// 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);
    }
}
/// $\forall a,\forall b,\forall c \in T, (a \circ b) \circ c = a \circ (b \circ c)$
pub trait Associative {}
/// associative binary operation
pub trait SemiGroup: Magma + Associative {}
impl<S: Magma + Associative> SemiGroup for S {}
/// $\exists e \in T, \forall a \in T, e \circ a = a \circ e = e$
pub trait Unital: Magma {
    /// 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();
    }
}
/// associative binary operation and an identity element
pub trait Monoid: SemiGroup + Unital {
    /// 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: SemiGroup + Unital> Monoid for M {}
/// $\exists e \in T, \forall a \in T, \exists b,c \in T, b \circ a = a \circ c = e$
pub trait Invertible: Magma {
    /// $a$ 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))
    }
}
/// associative binary operation and an identity element and inverse elements
pub trait Group: Monoid + Invertible {}
impl<G: Monoid + Invertible> Group for G {}
/// $\forall a,\forall b \in T, a \circ b = b \circ a$
pub trait Commutative {}
/// commutative monoid
pub trait AbelianMonoid: Monoid + Commutative {}
impl<M: Monoid + Commutative> AbelianMonoid for M {}
/// commutative group
pub trait AbelianGroup: Group + Commutative {}
impl<G: Group + Commutative> AbelianGroup for G {}
/// $\forall a \in T, a \circ a = a$
pub trait Idempotent {}
/// idempotent monoid
pub trait IdempotentMonoid: Monoid + Idempotent {}
impl<M: Monoid + Idempotent> IdempotentMonoid for M {}
#[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) ,*)) } ; }
/// $+$
pub struct AdditiveOperation<T: Clone + Zero + std::ops::Add<Output = T>> {
    _marker: std::marker::PhantomData<fn() -> T>,
}
mod additive_operation_impl {
    use super::*;
    use std::ops::{Add, Neg, Sub};
    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 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. });

mod tuple_operation_impl {
    #![allow(unused_variables)]
    use super::*;
    macro_rules ! impl_tuple_operation { ($ ($ M : ident) *, $ ($ i : tt) *) => { impl <$ ($ M : Magma) ,*> Magma for ($ ($ M ,) *) { type T = ($ (<$ M as Magma >:: T ,) *) ; # [inline] fn operate (x : & Self :: T , y : & Self :: T) -> Self :: T { ($ (<$ M as Magma >:: operate (& x .$ i , & y .$ i) ,) *) } } impl <$ ($ M : Unital) ,*> Unital for ($ ($ M ,) *) { # [inline] fn unit () -> Self :: T { ($ (<$ M as Unital >:: unit () ,) *) } } impl <$ ($ M : Associative) ,*> Associative for ($ ($ M ,) *) { } impl <$ ($ M : Commutative) ,*> Commutative for ($ ($ M ,) *) { } impl <$ ($ M : Idempotent) ,*> Idempotent for ($ ($ M ,) *) { } impl <$ ($ M : Invertible) ,*> Invertible for ($ ($ M ,) *) { # [inline] fn inverse (x : & Self :: T) -> Self :: T { ($ (<$ M as Invertible >:: inverse (& x .$ i) ,) *) } } } ; }
    impl_tuple_operation ! (,);
    impl_tuple_operation!(A, 0);
    impl_tuple_operation ! (A B , 0 1);
    impl_tuple_operation ! (A B C , 0 1 2);
    impl_tuple_operation ! (A B C D , 0 1 2 3);
    impl_tuple_operation ! (A B C D E , 0 1 2 3 4);
    impl_tuple_operation ! (A B C D E F , 0 1 2 3 4 5);
    impl_tuple_operation ! (A B C D E F G , 0 1 2 3 4 5 6);
    impl_tuple_operation ! (A B C D E F G H , 0 1 2 3 4 5 6 7);
    impl_tuple_operation ! (A B C D E F G H I , 0 1 2 3 4 5 6 7 8);
    impl_tuple_operation ! (A B C D E F G H I J , 0 1 2 3 4 5 6 7 8 9);
}
pub mod mint_basic {
    use super::*;
    #[macro_export]
    macro_rules ! define_basic_mintbase { ($ name : ident , $ m : expr , $ basety : ty , $ signedty : ty , $ upperty : ty , [$ ($ unsigned : ty) ,*] , [$ ($ signed : ty) ,*]) => { pub struct $ name ; impl MIntBase for $ name { type Inner = $ basety ; # [inline] fn get_mod () -> Self :: Inner { $ m } # [inline] fn mod_zero () -> Self :: Inner { 0 } # [inline] fn mod_one () -> Self :: Inner { 1 } # [inline] fn mod_add (x : Self :: Inner , y : Self :: Inner) -> Self :: Inner { let z = x + y ; let m = Self :: get_mod () ; if z >= m { z - m } else { z } } # [inline] fn mod_sub (x : Self :: Inner , y : Self :: Inner) -> Self :: Inner { if x < y { x + Self :: get_mod () - y } else { x - y } } # [inline] fn mod_mul (x : Self :: Inner , y : Self :: Inner) -> Self :: Inner { (x as $ upperty * y as $ upperty % Self :: get_mod () as $ upperty) as $ basety } # [inline] fn mod_div (x : Self :: Inner , y : Self :: Inner) -> Self :: Inner { Self :: mod_mul (x , Self :: mod_inv (y)) } # [inline] fn mod_neg (x : Self :: Inner) -> Self :: Inner { if x == 0 { 0 } else { Self :: get_mod () - x } } fn mod_inv (x : Self :: Inner) -> Self :: Inner { let p = Self :: get_mod () as $ signedty ; let (mut a , mut b) = (x as $ signedty , p) ; let (mut u , mut x) = (1 , 0) ; while a != 0 { let k = b / a ; x -= k * u ; b -= k * a ; std :: mem :: swap (& mut x , & mut u) ; std :: mem :: swap (& mut b , & mut a) ; } (if x < 0 { x + p } else { x }) as _ } } $ (impl MIntConvert <$ unsigned > for $ name { # [inline] fn from (x : $ unsigned) -> Self :: Inner { (x % < Self as MIntBase >:: get_mod () as $ unsigned) as $ basety } # [inline] fn into (x : Self :: Inner) -> $ unsigned { x as $ unsigned } # [inline] fn mod_into () -> $ unsigned { < Self as MIntBase >:: get_mod () as $ unsigned } }) * $ (impl MIntConvert <$ signed > for $ name { # [inline] fn from (x : $ signed) -> Self :: Inner { let x = x % < Self as MIntBase >:: get_mod () as $ signed ; if x < 0 { (x + < Self as MIntBase >:: get_mod () as $ signed) as $ basety } else { x as $ basety } } # [inline] fn into (x : Self :: Inner) -> $ signed { x as $ signed } # [inline] fn mod_into () -> $ signed { < Self as MIntBase >:: get_mod () as $ signed } }) * } ; }
    #[macro_export]
    macro_rules ! define_basic_mint32 { ($ ([$ name : ident , $ m : expr , $ mint_name : ident]) ,*) => { $ (crate :: define_basic_mintbase ! ($ name , $ m , u32 , i32 , u64 , [u32 , u64 , u128 , usize] , [i32 , i64 , i128 , isize]) ; pub type $ mint_name = MInt <$ name >;) * } ; }
    define_basic_mint32!(
        [Modulo998244353, 998_244_353, MInt998244353],
        [Modulo1000000007, 1_000_000_007, MInt1000000007],
        [Modulo1000000009, 1_000_000_009, MInt1000000009],
        [
            DynModuloU32,
            DYN_MODULUS_U32.with(|cell| unsafe { *cell.get() }),
            DynMIntU32
        ]
    );
    thread_local ! (static DYN_MODULUS_U32 : std :: cell :: UnsafeCell < u32 > = std :: cell :: UnsafeCell :: new (1_000_000_007));
    impl DynModuloU32 {
        pub fn set_mod(m: u32) {
            DYN_MODULUS_U32.with(|cell| unsafe { *cell.get() = m })
        }
    }
    thread_local ! (static DYN_MODULUS_U64 : std :: cell :: UnsafeCell < u64 > = std :: cell :: UnsafeCell :: new (1_000_000_007));
    define_basic_mintbase!(
        DynModuloU64,
        DYN_MODULUS_U64.with(|cell| unsafe { *cell.get() }),
        u64,
        i64,
        u128,
        [u64, u128, usize],
        [i64, i128, isize]
    );
    impl DynModuloU64 {
        pub fn set_mod(m: u64) {
            DYN_MODULUS_U64.with(|cell| unsafe { *cell.get() = m })
        }
    }
    pub type DynMIntU64 = MInt<DynModuloU64>;
    pub struct Modulo2;
    impl MIntBase for Modulo2 {
        type Inner = u32;
        #[inline]
        fn get_mod() -> Self::Inner {
            2
        }
        #[inline]
        fn mod_zero() -> Self::Inner {
            0
        }
        #[inline]
        fn mod_one() -> Self::Inner {
            1
        }
        #[inline]
        fn mod_add(x: Self::Inner, y: Self::Inner) -> Self::Inner {
            x ^ y
        }
        #[inline]
        fn mod_sub(x: Self::Inner, y: Self::Inner) -> Self::Inner {
            x ^ y
        }
        #[inline]
        fn mod_mul(x: Self::Inner, y: Self::Inner) -> Self::Inner {
            x | y
        }
        #[inline]
        fn mod_div(x: Self::Inner, y: Self::Inner) -> Self::Inner {
            assert_ne!(y, 0);
            x
        }
        #[inline]
        fn mod_neg(x: Self::Inner) -> Self::Inner {
            x
        }
        #[inline]
        fn mod_inv(x: Self::Inner) -> Self::Inner {
            assert_ne!(x, 0);
            x
        }
        #[inline]
        fn mod_pow(x: Self::Inner, y: usize) -> Self::Inner {
            if y == 0 {
                1
            } else {
                x
            }
        }
    }
    macro_rules ! impl_to_mint_base_for_modulo2 { ($ name : ident , $ basety : ty , [$ ($ t : ty) ,*]) => { $ (impl MIntConvert <$ t > for $ name { # [inline] fn from (x : $ t) -> Self :: Inner { (x & 1) as $ basety } # [inline] fn into (x : Self :: Inner) -> $ t { x as $ t } # [inline] fn mod_into () -> $ t { 1 } }) * } ; }
    impl_to_mint_base_for_modulo2!(
        Modulo2,
        u32,
        [u8, u16, u32, u64, u128, usize, i8, i16, i32, i64, i128, isize]
    );
    pub type MInt2 = MInt<Modulo2>;
}
#[repr(transparent)]
pub struct MInt<M>
where
    M: MIntBase,
{
    x: M::Inner,
    _marker: std::marker::PhantomData<fn() -> M>,
}
pub trait MIntBase {
    type Inner: Sized + Copy + Eq + std::fmt::Debug + std::hash::Hash;
    fn get_mod() -> Self::Inner;
    fn mod_zero() -> Self::Inner;
    fn mod_one() -> Self::Inner;
    fn mod_add(x: Self::Inner, y: Self::Inner) -> Self::Inner;
    fn mod_sub(x: Self::Inner, y: Self::Inner) -> Self::Inner;
    fn mod_mul(x: Self::Inner, y: Self::Inner) -> Self::Inner;
    fn mod_div(x: Self::Inner, y: Self::Inner) -> Self::Inner;
    fn mod_neg(x: Self::Inner) -> Self::Inner;
    fn mod_inv(x: Self::Inner) -> Self::Inner;
    fn mod_pow(x: Self::Inner, y: usize) -> Self::Inner {
        let (mut x, mut y, mut z) = (x, y, Self::mod_one());
        while y > 0 {
            if y & 1 == 1 {
                z = Self::mod_mul(z, x);
            }
            x = Self::mod_mul(x, x);
            y >>= 1;
        }
        z
    }
}
pub trait MIntConvert<T = <Self as MIntBase>::Inner>: MIntBase {
    fn from(x: T) -> <Self as MIntBase>::Inner;
    fn into(x: <Self as MIntBase>::Inner) -> T;
    fn mod_into() -> T;
}
mod mint_base {
    use super::*;
    use std::{
        fmt::{self, Debug, Display},
        hash::{Hash, Hasher},
        iter::{Product, Sum},
        marker::PhantomData,
        ops::{Add, AddAssign, Div, DivAssign, Mul, MulAssign, Neg, Sub, SubAssign},
        str::FromStr,
    };
    impl<M> MInt<M>
    where
        M: MIntConvert,
    {
        #[inline]
        pub fn new(x: M::Inner) -> Self {
            Self::new_unchecked(<M as MIntConvert<M::Inner>>::from(x))
        }
        #[inline]
        pub fn inner(self) -> M::Inner {
            <M as MIntConvert<M::Inner>>::into(self.x)
        }
    }
    impl<M> MInt<M>
    where
        M: MIntBase,
    {
        #[inline]
        pub fn new_unchecked(x: M::Inner) -> Self {
            Self {
                x,
                _marker: PhantomData,
            }
        }
        #[inline]
        pub fn get_mod() -> M::Inner {
            M::get_mod()
        }
        #[inline]
        pub fn pow(self, y: usize) -> Self {
            Self::new_unchecked(M::mod_pow(self.x, y))
        }
        #[inline]
        pub fn inv(self) -> Self {
            Self::new_unchecked(M::mod_inv(self.x))
        }
    }
    impl<M> Clone for MInt<M>
    where
        M: MIntBase,
    {
        #[inline]
        fn clone(&self) -> Self {
            Self {
                x: Clone::clone(&self.x),
                _marker: PhantomData,
            }
        }
    }
    impl<M> Copy for MInt<M> where M: MIntBase {}
    impl<M> Debug for MInt<M>
    where
        M: MIntBase,
    {
        fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
            Debug::fmt(&self.x, f)
        }
    }
    impl<M> Default for MInt<M>
    where
        M: MIntBase,
    {
        #[inline]
        fn default() -> Self {
            <Self as Zero>::zero()
        }
    }
    impl<M> PartialEq for MInt<M>
    where
        M: MIntBase,
    {
        #[inline]
        fn eq(&self, other: &Self) -> bool {
            PartialEq::eq(&self.x, &other.x)
        }
    }
    impl<M> Eq for MInt<M> where M: MIntBase {}
    impl<M> Hash for MInt<M>
    where
        M: MIntBase,
    {
        #[inline]
        fn hash<H: Hasher>(&self, state: &mut H) {
            Hash::hash(&self.x, state)
        }
    }
    macro_rules ! impl_mint_from { ($ ($ t : ty) ,*) => { $ (impl < M > From <$ t > for MInt < M > where M : MIntConvert <$ t >, { # [inline] fn from (x : $ t) -> Self { Self :: new_unchecked (< M as MIntConvert <$ t >>:: from (x)) } } impl < M > From < MInt < M >> for $ t where M : MIntConvert <$ t >, { # [inline] fn from (x : MInt < M >) -> $ t { < M as MIntConvert <$ t >>:: into (x . x) } }) * } ; }
    impl_mint_from!(u8, u16, u32, u64, u128, usize, i8, i16, i32, i64, i128, isize);
    impl<M> Zero for MInt<M>
    where
        M: MIntBase,
    {
        #[inline]
        fn zero() -> Self {
            Self::new_unchecked(M::mod_zero())
        }
    }
    impl<M> One for MInt<M>
    where
        M: MIntBase,
    {
        #[inline]
        fn one() -> Self {
            Self::new_unchecked(M::mod_one())
        }
    }
    impl<M> Add for MInt<M>
    where
        M: MIntBase,
    {
        type Output = Self;
        #[inline]
        fn add(self, rhs: Self) -> Self::Output {
            Self::new_unchecked(M::mod_add(self.x, rhs.x))
        }
    }
    impl<M> Sub for MInt<M>
    where
        M: MIntBase,
    {
        type Output = Self;
        #[inline]
        fn sub(self, rhs: Self) -> Self::Output {
            Self::new_unchecked(M::mod_sub(self.x, rhs.x))
        }
    }
    impl<M> Mul for MInt<M>
    where
        M: MIntBase,
    {
        type Output = Self;
        #[inline]
        fn mul(self, rhs: Self) -> Self::Output {
            Self::new_unchecked(M::mod_mul(self.x, rhs.x))
        }
    }
    impl<M> Div for MInt<M>
    where
        M: MIntBase,
    {
        type Output = Self;
        #[inline]
        fn div(self, rhs: Self) -> Self::Output {
            Self::new_unchecked(M::mod_div(self.x, rhs.x))
        }
    }
    impl<M> Neg for MInt<M>
    where
        M: MIntBase,
    {
        type Output = Self;
        #[inline]
        fn neg(self) -> Self::Output {
            Self::new_unchecked(M::mod_neg(self.x))
        }
    }
    impl<M> Sum for MInt<M>
    where
        M: MIntBase,
    {
        #[inline]
        fn sum<I: Iterator<Item = Self>>(iter: I) -> Self {
            iter.fold(<Self as Zero>::zero(), Add::add)
        }
    }
    impl<M> Product for MInt<M>
    where
        M: MIntBase,
    {
        #[inline]
        fn product<I: Iterator<Item = Self>>(iter: I) -> Self {
            iter.fold(<Self as One>::one(), Mul::mul)
        }
    }
    impl<'a, M: 'a> Sum<&'a MInt<M>> for MInt<M>
    where
        M: MIntBase,
    {
        #[inline]
        fn sum<I: Iterator<Item = &'a Self>>(iter: I) -> Self {
            iter.fold(<Self as Zero>::zero(), Add::add)
        }
    }
    impl<'a, M: 'a> Product<&'a MInt<M>> for MInt<M>
    where
        M: MIntBase,
    {
        #[inline]
        fn product<I: Iterator<Item = &'a Self>>(iter: I) -> Self {
            iter.fold(<Self as One>::one(), Mul::mul)
        }
    }
    impl<M> Display for MInt<M>
    where
        M: MIntConvert,
        M::Inner: Display,
    {
        fn fmt<'a>(&self, f: &mut fmt::Formatter<'a>) -> Result<(), fmt::Error> {
            write!(f, "{}", self.inner())
        }
    }
    impl<M> FromStr for MInt<M>
    where
        M: MIntConvert,
        M::Inner: FromStr,
    {
        type Err = <M::Inner as FromStr>::Err;
        #[inline]
        fn from_str(s: &str) -> Result<Self, Self::Err> {
            s.parse::<M::Inner>().map(Self::new)
        }
    }
    impl<M> IterScan for MInt<M>
    where
        M: MIntConvert,
        M::Inner: FromStr,
    {
        type Output = Self;
        #[inline]
        fn scan<'a, I: Iterator<Item = &'a str>>(iter: &mut I) -> Option<Self::Output> {
            iter.next()?.parse::<MInt<M>>().ok()
        }
    }
    macro_rules! impl_mint_ref_binop {
        ($ imp : ident , $ method : ident , $ t : ty) => {
            impl<M> $imp<$t> for &$t
            where
                M: MIntBase,
            {
                type Output = <$t as $imp<$t>>::Output;
                #[inline]
                fn $method(self, other: $t) -> <$t as $imp<$t>>::Output {
                    $imp::$method(*self, other)
                }
            }
            impl<M> $imp<&$t> for $t
            where
                M: MIntBase,
            {
                type Output = <$t as $imp<$t>>::Output;
                #[inline]
                fn $method(self, other: &$t) -> <$t as $imp<$t>>::Output {
                    $imp::$method(self, *other)
                }
            }
            impl<M> $imp<&$t> for &$t
            where
                M: MIntBase,
            {
                type Output = <$t as $imp<$t>>::Output;
                #[inline]
                fn $method(self, other: &$t) -> <$t as $imp<$t>>::Output {
                    $imp::$method(*self, *other)
                }
            }
        };
    }
    impl_mint_ref_binop!(Add, add, MInt<M>);
    impl_mint_ref_binop!(Sub, sub, MInt<M>);
    impl_mint_ref_binop!(Mul, mul, MInt<M>);
    impl_mint_ref_binop!(Div, div, MInt<M>);
    macro_rules! impl_mint_ref_unop {
        ($ imp : ident , $ method : ident , $ t : ty) => {
            impl<M> $imp for &$t
            where
                M: MIntBase,
            {
                type Output = <$t as $imp>::Output;
                #[inline]
                fn $method(self) -> <$t as $imp>::Output {
                    $imp::$method(*self)
                }
            }
        };
    }
    impl_mint_ref_unop!(Neg, neg, MInt<M>);
    macro_rules! impl_mint_ref_op_assign {
        ($ imp : ident , $ method : ident , $ t : ty , $ fromimp : ident , $ frommethod : ident) => {
            impl<M> $imp<$t> for $t
            where
                M: MIntBase,
            {
                #[inline]
                fn $method(&mut self, rhs: $t) {
                    *self = $fromimp::$frommethod(*self, rhs);
                }
            }
            impl<M> $imp<&$t> for $t
            where
                M: MIntBase,
            {
                #[inline]
                fn $method(&mut self, other: &$t) {
                    $imp::$method(self, *other);
                }
            }
        };
    }
    impl_mint_ref_op_assign!(AddAssign, add_assign, MInt<M>, Add, add);
    impl_mint_ref_op_assign!(SubAssign, sub_assign, MInt<M>, Sub, sub);
    impl_mint_ref_op_assign!(MulAssign, mul_assign, MInt<M>, Mul, mul);
    impl_mint_ref_op_assign!(DivAssign, div_assign, MInt<M>, Div, div);
}