// ---------- begin ModInt ---------- mod modint { #[allow(dead_code)] pub struct Mod; impl ConstantModulo for Mod { const MOD: u32 = 998_244_353; } #[allow(dead_code)] pub struct StaticMod; static mut STATIC_MOD: u32 = 0; impl Modulo for StaticMod { fn modulo() -> u32 { unsafe { STATIC_MOD } } } #[allow(dead_code)] impl StaticMod { pub fn set_modulo(p: u32) { unsafe { STATIC_MOD = p; } } } use std::marker::*; use std::ops::*; pub trait Modulo { fn modulo() -> u32; } pub trait ConstantModulo { const MOD: u32; } impl Modulo for T where T: ConstantModulo, { fn modulo() -> u32 { T::MOD } } pub struct ModInt(pub u32, PhantomData); impl Clone for ModInt { fn clone(&self) -> Self { ModInt::new_unchecked(self.0) } } impl Copy for ModInt {} impl Add for ModInt { type Output = ModInt; fn add(self, rhs: Self) -> Self::Output { let mut d = self.0 + rhs.0; if d >= T::modulo() { d -= T::modulo(); } ModInt::new_unchecked(d) } } impl AddAssign for ModInt { fn add_assign(&mut self, rhs: Self) { *self = *self + rhs; } } impl Sub for ModInt { type Output = ModInt; fn sub(self, rhs: Self) -> Self::Output { let mut d = T::modulo() + self.0 - rhs.0; if d >= T::modulo() { d -= T::modulo(); } ModInt::new_unchecked(d) } } impl SubAssign for ModInt { fn sub_assign(&mut self, rhs: Self) { *self = *self - rhs; } } impl Mul for ModInt { type Output = ModInt; fn mul(self, rhs: Self) -> Self::Output { let v = self.0 as u64 * rhs.0 as u64 % T::modulo() as u64; ModInt::new_unchecked(v as u32) } } impl MulAssign for ModInt { fn mul_assign(&mut self, rhs: Self) { *self = *self * rhs; } } impl Neg for ModInt { type Output = ModInt; fn neg(self) -> Self::Output { if self.0 == 0 { Self::zero() } else { Self::new_unchecked(T::modulo() - self.0) } } } impl std::fmt::Display for ModInt { fn fmt<'a>(&self, f: &mut std::fmt::Formatter<'a>) -> std::fmt::Result { write!(f, "{}", self.0) } } impl std::fmt::Debug for ModInt { fn fmt<'a>(&self, f: &mut std::fmt::Formatter<'a>) -> std::fmt::Result { write!(f, "{}", self.0) } } impl std::str::FromStr for ModInt { type Err = std::num::ParseIntError; fn from_str(s: &str) -> Result { let val = s.parse::()?; Ok(ModInt::new(val)) } } impl From for ModInt { fn from(val: usize) -> ModInt { ModInt::new_unchecked((val % T::modulo() as usize) as u32) } } impl From for ModInt { fn from(val: u64) -> ModInt { ModInt::new_unchecked((val % T::modulo() as u64) as u32) } } impl From for ModInt { fn from(val: i64) -> ModInt { let m = T::modulo() as i64; ModInt::new((val % m + m) as u32) } } #[allow(dead_code)] impl ModInt { pub fn new_unchecked(d: u32) -> Self { ModInt(d, PhantomData) } pub fn zero() -> Self { ModInt::new_unchecked(0) } pub fn one() -> Self { ModInt::new_unchecked(1) } pub fn is_zero(&self) -> bool { self.0 == 0 } } #[allow(dead_code)] impl ModInt { pub fn new(d: u32) -> Self { ModInt::new_unchecked(d % T::modulo()) } pub fn pow(&self, mut n: u64) -> Self { let mut t = Self::one(); let mut s = *self; while n > 0 { if n & 1 == 1 { t *= s; } s *= s; n >>= 1; } t } pub fn inv(&self) -> Self { assert!(self.0 != 0); self.pow(T::modulo() as u64 - 2) } } } // ---------- end ModInt ---------- // ---------- begin Precalc ---------- mod precalc { use super::modint::*; #[allow(dead_code)] pub struct Precalc { inv: Vec>, fact: Vec>, ifact: Vec>, } #[allow(dead_code)] impl Precalc { pub fn new(n: usize) -> Precalc { let mut inv = vec![ModInt::one(); n + 1]; let mut fact = vec![ModInt::one(); n + 1]; let mut ifact = vec![ModInt::one(); n + 1]; for i in 2..(n + 1) { fact[i] = fact[i - 1] * ModInt::new_unchecked(i as u32); } ifact[n] = fact[n].inv(); if n > 0 { inv[n] = ifact[n] * fact[n - 1]; } for i in (1..n).rev() { ifact[i] = ifact[i + 1] * ModInt::new_unchecked((i + 1) as u32); inv[i] = ifact[i] * fact[i - 1]; } Precalc { inv: inv, fact: fact, ifact: ifact, } } pub fn inv(&self, n: usize) -> ModInt { assert!(n > 0); self.inv[n] } pub fn fact(&self, n: usize) -> ModInt { self.fact[n] } pub fn ifact(&self, n: usize) -> ModInt { self.ifact[n] } pub fn perm(&self, n: usize, k: usize) -> ModInt { if k > n { return ModInt::zero(); } self.fact[n] * self.ifact[n - k] } pub fn comb(&self, n: usize, k: usize) -> ModInt { if k > n { return ModInt::zero(); } self.fact[n] * self.ifact[k] * self.ifact[n - k] } } } // ---------- end Precalc ---------- use modint::*; type M = ModInt; // ---------- begin Matrix ---------- mod matrix { use std::ops::*; pub trait SemiRing: Add + Mul + Copy { fn zero() -> Self; fn one() -> Self; } pub trait Inverse: SemiRing { fn is_zero() -> bool; fn inv(self) -> Self; } #[derive(Clone)] pub struct SquareMatrix { size: usize, buf: Box<[R]>, } #[allow(dead_code)] impl SquareMatrix { pub fn zero(size: usize) -> Self { SquareMatrix { size: size, buf: vec![R::zero(); size * size].into_boxed_slice(), } } pub fn identity(size: usize) -> Self { let mut e = Self::zero(size); for i in 0..size { e.buf[i * size + i] = R::one(); } e } pub fn set_at(&mut self, x: usize, y: usize, val: R) { assert!(x < self.size && y < self.size); self.buf[x * self.size + y] = val; } pub fn get_at(&self, x: usize, y: usize) -> R { assert!(x < self.size && y < self.size); self.buf[x * self.size + y] } pub fn get_mut(&mut self, x: usize, y: usize) -> &mut R { assert!(x < self.size && y < self.size); &mut self.buf[x * self.size + y] } pub fn matadd(&self, rhs: &Self) -> Self { assert!(self.size == rhs.size); let buf: Vec = self .buf .iter() .zip(rhs.buf.iter()) .map(|p| *p.0 + *p.1) .collect(); SquareMatrix { size: self.size, buf: buf.into_boxed_slice(), } } pub fn matmul(&self, rhs: &Self) -> Self { let size = self.size; assert!(size == rhs.size); let mut res = Self::zero(size); for (x, a) in res.buf.chunks_mut(size).zip(self.buf.chunks(size)) { for (a, b) in a.iter().zip(rhs.buf.chunks(size)) { for (x, b) in x.iter_mut().zip(b.iter()) { *x = *x + *a * *b; } } } res } pub fn mat_pow(&self, mut n: usize) -> Self { let size = self.size; let mut t = Self::identity(size); let mut s = self.clone(); while n > 0 { if n & 1 == 1 { t = t.matmul(&s); } s = s.matmul(&s); n >>= 1; } t } } #[allow(dead_code)] impl> SquareMatrix { pub fn matsub(&self, rhs: &Self) -> Self { assert!(self.size == rhs.size); let buf: Vec = self .buf .iter() .zip(rhs.buf.iter()) .map(|p| *p.0 - *p.1) .collect(); SquareMatrix { size: self.size, buf: buf.into_boxed_slice(), } } } /* #[allow(dead_code)] impl> SquareMatrix { pub fn inverse(&self) -> Self { } } */ } // ---------- end Matrix ---------- // ---------- begin input macro ---------- // reference: https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8 macro_rules! input { (source = $s:expr, $($r:tt)*) => { let mut iter = $s.split_whitespace(); input_inner!{iter, $($r)*} }; ($($r:tt)*) => { let s = { use std::io::Read; let mut s = String::new(); std::io::stdin().read_to_string(&mut s).unwrap(); s }; let mut iter = s.split_whitespace(); input_inner!{iter, $($r)*} }; } macro_rules! input_inner { ($iter:expr) => {}; ($iter:expr, ) => {}; ($iter:expr, $var:ident : $t:tt $($r:tt)*) => { let $var = read_value!($iter, $t); input_inner!{$iter $($r)*} }; } macro_rules! read_value { ($iter:expr, ( $($t:tt),* )) => { ( $(read_value!($iter, $t)),* ) }; ($iter:expr, [ $t:tt ; $len:expr ]) => { (0..$len).map(|_| read_value!($iter, $t)).collect::>() }; ($iter:expr, chars) => { read_value!($iter, String).chars().collect::>() }; ($iter:expr, bytes) => { read_value!($iter, String).bytes().collect::>() }; ($iter:expr, usize1) => { read_value!($iter, usize) - 1 }; ($iter:expr, $t:ty) => { $iter.next().unwrap().parse::<$t>().expect("Parse error") }; } // ---------- end input macro ---------- use matrix::*; impl SemiRing for M { fn zero() -> Self { M::zero() } fn one() -> Self { M::one() } } type Mat = SquareMatrix; fn run() { input! { a: (M, M, usize), b: (M, M, usize), k: usize, } let p = a.0 * a.1.inv(); let q = b.0 * b.1.inv(); let size = a.2 + b.2 + 1; let mut trans = Mat::identity(size); let mut mat = Mat::zero(size); mat.set_at(size - 1, size - 1, M::one()); mat.set_at(0, 0, M::one()); for src in 1..(size - 1) { let mut prob = M::one(); for dst in src..size { mat.set_at(src, dst, prob * (M::one() - p)); prob *= p; } *mat.get_mut(src, size - 1) += prob; } trans = trans.matmul(&mat); let mut mat = Mat::zero(size); mat.set_at(size - 1, size - 1, M::one()); mat.set_at(0, 0, M::one()); for src in 1..(size - 1) { let mut prob = M::one(); for dst in (0..=src).rev() { mat.set_at(src, dst, prob * (M::one() - q)); prob *= q; } *mat.get_mut(src, 0) += prob; } trans = trans.matmul(&mat); let pow = trans.mat_pow(k); println!("{}\n{}", pow.get_at(b.2, size - 1), pow.get_at(b.2, 0)); } fn main() { run(); }