Problem:
Score : 700 points

Problem Statement

We have a graph with N vertices and M edges, and there are two people on the graph: Takahashi and Aoki.

The i-th edge connects Vertex U_i and Vertex V_i.
The time it takes to traverse this edge is D_i minutes, regardless of direction and who traverses the edge (Takahashi or Aoki).

Takahashi departs Vertex S and Aoki departs Vertex T at the same time. Takahashi travels to Vertex T and Aoki travels to Vertex S, both in the shortest time possible.
Find the number of the pairs of ways for Takahashi and Aoki to choose their shortest paths such that they never meet (at a vertex or on an edge) during the travel, modulo 10^9 + 7.

Constraints

1 \leq N \leq 100 000

1 \leq M \leq 200 000

1 \leq S, T \leq N

S \neq T

1 \leq U_i, V_i \leq N (1 \leq i \leq M)

1 \leq D_i \leq 10^9 (1 \leq i \leq M)

If i \neq j, then (U_i, V_i) \neq (U_j, V_j) and (U_i, V_i) \neq (V_j, U_j).

U_i \neq V_i (1 \leq i \leq M)

D_i are integers.

The given graph is connected.

Input

Input is given from Standard Input in the following format:

N M
S T
U_1 V_1 D_1
U_2 V_2 D_2
:
U_M V_M D_M

Output

Print the answer.

Sample Input 1

4 4
1 3
1 2 1
2 3 1
3 4 1
4 1 1

Sample Output 1

2

There are two ways to choose shortest paths that satisfies the condition:

Takahashi chooses the path 1 \rightarrow 2 \rightarrow 3, and Aoki chooses the path 3 \rightarrow 4 \rightarrow 1.

Takahashi chooses the path 1 \rightarrow 4 \rightarrow 3, and Aoki chooses the path 3 \rightarrow 2 \rightarrow 1.

Sample Input 2

3 3
1 3
1 2 1
2 3 1
3 1 2

Sample Output 2

2

Sample Input 3

3 3
1 3
1 2 1
2 3 1
3 1 2

Sample Output 3

2

Sample Input 4

8 13
4 2
7 3 9
6 2 3
1 6 4
7 6 9
3 8 9
1 2 2
2 8 12
8 6 9
2 5 5
4 2 18
5 3 7
5 1 515371567
4 8 6

Sample Output 4

6

Solution in Rust (1.42.0) (judge verdict: Accepted)

use proconio::{input, marker::Usize1};
use std::cmp::Reverse;

// modint {{{
mod modint {
    pub trait ModValue:
        std::ops::Add<Output = Self>
        + std::ops::Sub<Output = Self>
        + std::ops::Mul<Output = Self>
        + std::ops::Div<Output = Self>
        + std::ops::Rem<Output = Self>
        + std::ops::AddAssign
        + std::ops::SubAssign
        + std::ops::MulAssign
        + std::ops::DivAssign
        + std::ops::RemAssign
        + std::ops::Neg
        + std::cmp::Ord
        + Clone
        + Copy
        + Sized
        + std::fmt::Debug
        + std::fmt::Display
    {
    }

    impl<
            Value: std::ops::Add<Output = Self>
                + std::ops::Sub<Output = Self>
                + std::ops::Mul<Output = Self>
                + std::ops::Div<Output = Self>
                + std::ops::Rem<Output = Self>
                + std::ops::AddAssign
                + std::ops::SubAssign
                + std::ops::MulAssign
                + std::ops::DivAssign
                + std::ops::RemAssign
                + std::ops::Neg
                + std::cmp::Ord
                + Clone
                + Copy
                + Sized
                + std::fmt::Debug
                + std::fmt::Display,
        > ModValue for Value
    {
    }

    pub trait Minfo: Clone + Copy + std::fmt::Debug {
        type Value: ModValue;

        fn modulus() -> Self::Value;

        fn zero() -> Self::Value;

        fn one() -> Self::Value;
    }

    #[derive(Clone, Copy, Debug)]
    pub struct Mint<Mod: Minfo>(Mod::Value);

    impl<Mod: Minfo> Mint<Mod> {
        #[inline]
        #[allow(dead_code)]
        fn is_within_the_range(x: Mod::Value) -> bool {
            Mod::zero() <= x && x < Mod::modulus()
        }

        #[inline]
        #[allow(dead_code)]
        fn normalize(x: Mod::Value) -> Mod::Value {
            if Self::is_within_the_range(x) {
                x
            } else {
                let x = x % Mod::modulus();
                if x < Mod::zero() {
                    x + Mod::modulus()
                } else {
                    x
                }
            }
        }

        #[allow(dead_code)]
        pub fn zero() -> Self {
            Self(Mod::zero())
        }

        #[allow(dead_code)]
        pub fn one() -> Self {
            Self(Mod::one())
        }

        #[allow(dead_code)]
        pub fn modulus() -> Mod::Value {
            Mod::modulus()
        }

        #[allow(dead_code)]
        pub fn is_zero(&self) -> bool {
            self.value() == Mod::zero()
        }

        #[allow(dead_code)]
        pub fn from_value(x: Mod::Value) -> Self {
            Self(Self::normalize(x))
        }

        #[allow(dead_code)]
        pub fn value(&self) -> Mod::Value {
            self.0
        }

        #[allow(dead_code)]
        pub fn inv(&self) -> Self {
            if self.is_zero() {
                panic!("inv(0)");
            }
            Self(Self::raw_inv(self.value()))
        }

        #[allow(dead_code)]
        pub fn pow(&self, b: u64) -> Self {
            Self(Self::raw_pow(self.value(), b))
        }

        fn raw_add(x: Mod::Value, y: Mod::Value) -> Mod::Value {
            let z = x + y;
            if Mod::modulus() <= z {
                z - Mod::modulus()
            } else {
                z
            }
        }

        fn raw_sub(x: Mod::Value, y: Mod::Value) -> Mod::Value {
            let z = x - y;
            if z < Mod::zero() {
                z + Mod::modulus()
            } else {
                z
            }
        }

        fn raw_neg(x: Mod::Value) -> Mod::Value {
            if x == Mod::zero() {
                Mod::zero()
            } else {
                Mod::modulus() - x
            }
        }

        fn raw_mul(x: Mod::Value, y: Mod::Value) -> Mod::Value {
            x * y % Mod::modulus()
        }

        fn raw_div(x: Mod::Value, y: Mod::Value) -> Mod::Value {
            Self::raw_mul(x, Self::raw_inv(y))
        }

        #[allow(clippy::many_single_char_names)]
        fn raw_inv(mut x: Mod::Value) -> Mod::Value {
            let mut y = Mod::modulus();
            let mut u = Mod::one();
            let mut v = Mod::zero();
            while x != Mod::zero() {
                let q = y / x;
                y -= x * q;
                v -= u * q;
                std::mem::swap(&mut x, &mut y);
                std::mem::swap(&mut u, &mut v);
            }
            Self::normalize(v)
        }

        fn raw_pow(mut a: Mod::Value, mut b: u64) -> Mod::Value {
            let mut x = Mod::one();
            while 0 < b {
                if b % 2 == 1 {
                    x = Self::raw_mul(x, a);
                }
                a = Self::raw_mul(a, a);
                b /= 2
            }
            x
        }
    }

    macro_rules! impl_biop {
        (
            $(
                $biop_trait: ident::$biop_fn: ident,
                $biop_assign_trait: ident::$biop_assign_fn: ident,
                $biop_impl: ident;
            )*
        ) => {
            $(
                impl<Mod: Minfo> std::ops::$biop_trait for Mint<Mod> {
                    type Output = Self;
                    fn $biop_fn(self, rhs: Self) -> Self::Output {
                        Mint(Self::$biop_impl(self.value(), rhs.value()))
                    }
                }
                impl<Mod: Minfo> std::ops::$biop_assign_trait for Mint<Mod>
                where
                    Self: Copy
                {
                    fn $biop_assign_fn(&mut self, rhs: Self) {
                        use std::ops::$biop_trait;
                        *self = Mint::$biop_fn(*self, rhs)
                    }
                }
            )*
        };
    }

    impl_biop! {
        Add::add, AddAssign::add_assign, raw_add;
        Sub::sub, SubAssign::sub_assign, raw_sub;
        Mul::mul, MulAssign::mul_assign, raw_mul;
        Div::div, DivAssign::div_assign, raw_div;
    }

    impl<Mod: Minfo> std::ops::Neg for Mint<Mod> {
        type Output = Self;
        fn neg(self) -> Self::Output {
            Self(Self::raw_neg(self.value()))
        }
    }

    impl<Mod: Minfo> std::cmp::PartialEq for Mint<Mod> {
        fn eq(&self, other: &Self) -> bool {
            self.value().eq(&other.value())
        }
    }
    impl<Mod: Minfo> std::cmp::Eq for Mint<Mod> {}

    impl<Mod: Minfo> std::fmt::Display for Mint<Mod> {
        fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
            write!(f, "{}", self.value())
        }
    }

    impl<Mod: Minfo> std::iter::Sum<Self> for Mint<Mod> {
        fn sum<I: Iterator<Item = Self>>(iter: I) -> Self {
            iter.fold(Mint::zero(), |acc, x| acc + x)
        }
    }

    impl<Mod: Minfo> std::iter::Product<Self> for Mint<Mod> {
        fn product<I: Iterator<Item = Self>>(iter: I) -> Self {
            iter.fold(Mint::one(), |acc, x| acc * x)
        }
    }

    #[allow(dead_code)]
    pub struct Factorial<Mod: Minfo> {
        normal: Vec<Mint<Mod>>,
        inverted: Vec<Mint<Mod>>,
    }

    #[allow(dead_code)]
    impl<Mod: Minfo> Factorial<Mod>
    where
        Mod::Value: std::convert::TryFrom<usize, Error = std::num::TryFromIntError>,
    {
        pub fn new(n: usize) -> Self {
            if n == 0 {
                Self {
                    normal: Vec::new(),
                    inverted: Vec::new(),
                }
            } else {
                use std::convert::TryFrom;
                let mut normal = vec![Mint::one(); n];
                for i in 1..n {
                    normal[i] = normal[i - 1] * Mint::from_value(Mod::Value::try_from(i).unwrap());
                }
                let mut inverted = vec![Mint::one(); n];
                inverted[n - 1] = normal[n - 1].inv();
                for i in (1..n).rev() {
                    inverted[i - 1] =
                        inverted[i] * Mint::from_value(Mod::Value::try_from(i).unwrap());
                }
                Self { normal, inverted }
            }
        }
    }

    #[allow(dead_code)]
    impl<Mod: Minfo> Factorial<Mod> {
        pub fn is_empty(&self) -> bool {
            self.normal.is_empty()
        }

        pub fn len(&self) -> usize {
            self.normal.len()
        }

        pub fn inverted(&self, i: usize) -> Mint<Mod> {
            self.inverted[i]
        }

        pub fn falling(&self, x: usize, p: usize) -> Mint<Mod> {
            if p == 0 {
                Mint::one()
            } else if x < p {
                Mint::zero()
            } else {
                assert!(x < self.len());
                self[x] * self.inverted(x - p)
            }
        }

        pub fn binom(&self, n: usize, k: usize) -> Mint<Mod> {
            if n < k {
                Mint::zero()
            } else {
                assert!(n < self.len());
                self[n] * self.inverted(k) * self.inverted(n - k)
            }
        }
    }

    #[allow(dead_code)]
    impl<Mod: Minfo, I: std::slice::SliceIndex<[Mint<Mod>]>> std::ops::Index<I> for Factorial<Mod> {
        type Output = I::Output;

        #[inline]
        fn index(&self, index: I) -> &Self::Output {
            &self.normal[index]
        }
    }

    #[allow(dead_code)]
    pub struct DoubleFactorial<Mod: Minfo> {
        normal: Vec<Mint<Mod>>,
        inverted: Vec<Mint<Mod>>,
    }

    #[allow(dead_code)]
    impl<Mod: Minfo> DoubleFactorial<Mod>
    where
        Mod::Value: std::convert::TryFrom<usize, Error = std::num::TryFromIntError>,
    {
        pub fn new(n: usize) -> Self {
            use std::convert::TryFrom;
            let mut normal = vec![Mint::one(); n];
            for i in 2..n {
                normal[i] = normal[i - 2] * Mint::from_value(Mod::Value::try_from(i).unwrap());
            }
            let mut inverted = vec![Mint::one(); n];
            inverted[n - 1] = normal[n - 1].inv();
            inverted[n - 2] = normal[n - 2].inv();
            for i in (2..n).rev() {
                inverted[i - 2] = inverted[i] * Mint::from_value(Mod::Value::try_from(i).unwrap());
            }
            Self { normal, inverted }
        }
    }

    #[allow(dead_code)]
    impl<Mod: Minfo> DoubleFactorial<Mod> {
        pub fn inverted(&self, i: usize) -> Mint<Mod> {
            self.inverted[i]
        }
    }

    #[allow(dead_code)]
    impl<Mod: Minfo, I: std::slice::SliceIndex<[Mint<Mod>]>> std::ops::Index<I>
        for DoubleFactorial<Mod>
    {
        type Output = I::Output;

        #[inline]
        fn index(&self, index: I) -> &Self::Output {
            &self.normal[index]
        }
    }

    #[macro_export]
    macro_rules! define_mint {
        (
            struct $modint: ident ($modinfo: ident ($modvalue: expr; $modty: ty));
        ) => {
            #[allow(dead_code)]
            #[derive(Clone, Copy, Debug)]
            pub struct $modinfo {}
            impl Minfo for $modinfo {
                type Value = $modty;
                fn modulus() -> Self::Value {
                    $modvalue
                }
                fn zero() -> Self::Value {
                    0
                }
                fn one() -> Self::Value {
                    1
                }
            }
            type $modint = Mint<$modinfo>;
        };
    }

    #[allow(dead_code)]
    #[derive(Clone, Copy, Debug)]
    pub struct Minfo1000000007 {}

    impl Minfo for Minfo1000000007 {
        type Value = i64;
        fn modulus() -> Self::Value {
            1_000_000_007
        }
        fn zero() -> Self::Value {
            0
        }
        fn one() -> Self::Value {
            1
        }
    }

    #[allow(dead_code)]
    pub type Mint1000000007 = Mint<Minfo1000000007>;

    #[allow(dead_code)]
    #[derive(Clone, Copy, Debug)]
    pub struct Minfo998244353 {}

    impl Minfo for Minfo998244353 {
        type Value = i64;
        fn modulus() -> Self::Value {
            998_244_353
        }
        fn zero() -> Self::Value {
            0
        }
        fn one() -> Self::Value {
            1
        }
    }

    #[allow(dead_code)]
    pub type Mint998244353 = Mint<Minfo998244353>;
}
// }}}

type Mint = modint::Mint1000000007;

fn main() {
    input!(
        n: usize,
        m: usize,
        s: Usize1,
        t: Usize1,
        uvd: [(Usize1, Usize1, u64); m]
    );

    let mut g = vec![vec![]; n];
    for &(u, v, d) in &uvd {
        g[u].push((v, d));
        g[v].push((u, d));
    }

    let mut heap = std::collections::BinaryHeap::from(vec![(Reverse(0), s)]);
    let mut dist = vec![std::u64::MAX; n];
    dist[s] = 0;
    while let Some((Reverse(dx), x)) = heap.pop() {
        if dist[x] < dx {
            continue;
        }
        for &(y, w) in &g[x] {
            let dy = dx + w;
            if dist[y] <= dy {
                continue;
            }
            dist[y] = dy;
            heap.push((Reverse(dy), y));
        }
    }
    let dist_s2t = dist[t];

    let mut ord: Vec<usize> = (0..n).collect();
    ord.sort_by_key(|&i| dist[i]);
    let mut on_shortest = vec![false; n];
    on_shortest[t] = true;
    let mut dp = vec![Mint::zero(); n];
    let mut ep = vec![Mint::zero(); n];
    dp[s] = Mint::one();
    ep[t] = Mint::one();
    for &x in &ord {
        for &(y, d) in &g[x] {
            if dist[x] + d == dist[y] {
                let dpx = dp[x];
                dp[y] += dpx;
            }
        }
    }
    for x in ord.iter().copied().rev() {
        for &(y, d) in &g[x] {
            if dist[x] == dist[y] + d {
                let epx = ep[x];
                ep[y] += epx;
                on_shortest[y] |= on_shortest[x];
            }
        }
    }

    println!(
        "{}",
        dp[t].pow(2)
            - dist
                .iter()
                .copied()
                .enumerate()
                .filter_map(|(i, d)| if on_shortest[i] && 2 * d == dist_s2t {
                    Some(dp[i] * ep[i])
                } else {
                    None
                })
                .chain(uvd.iter().filter_map(|&(u, v, w)| {
                    let du = dist[u];
                    let dv = dist[v];
                    if on_shortest[u] && on_shortest[v] && du.max(dv) - du.min(dv) == w {
                        if 2 * du < dist_s2t && dist_s2t < 2 * dv {
                            Some(dp[u] * ep[v])
                        } else if 2 * dv < dist_s2t && dist_s2t < 2 * du {
                            Some(dp[v] * ep[u])
                        } else {
                            None
                        }
                    } else {
                        None
                    }
                }))
                .map(|x| x * x)
                .sum::<Mint>()
    );
}