Problem:
Score : 600 points

Problem Statement

Snuke is standing on a two-dimensional plane. In one operation, he can move by 1 in the positive x-direction, or move by 1 in the positive y-direction.

Let us define a function f(r, c) as follows:

f(r,c) := (The number of paths from the point (0, 0) to the point (r, c) that Snuke can trace by repeating the operation above)

Given are integers r_1, r_2, c_1, and c_2.
Find the sum of f(i, j) over all pair of integers (i, j) such that r_1 ≤ i ≤ r_2 and c_1 ≤ j ≤ c_2, and compute this value modulo (10^9+7).

Constraints

1 ≤ r_1 ≤ r_2 ≤ 10^6

1 ≤ c_1 ≤ c_2 ≤ 10^6

All values in input are integers.

Input

Input is given from Standard Input in the following format:

r_1 c_1 r_2 c_2

Output

Print the sum of f(i, j) modulo (10^9+7).

Sample Input 1

1 1 2 2

Sample Output 1

14

For example, there are two paths from the point (0, 0) to the point (1, 1): (0,0) → (0,1) → (1,1) and (0,0) → (1,0) → (1,1), so f(1,1)=2.

Similarly, f(1,2)=3, f(2,1)=3, and f(2,2)=6. Thus, the sum is 14.

Sample Input 2

314 159 2653 589

Sample Output 2

602215194

Solution in PyPy3 (2.4.0) (judge verdict: Accepted)

class Calc:
    def __init__(self, max_value, mod):
        """combination(max_value, all)"""
        fact = [-1] * (max_value + 1)
        fact[0] = 1
        fact[1] = 1
        for x in range(2, max_value + 1):
            fact[x] = x * fact[x - 1] % mod

        invs = [1] * (max_value + 1)
        invs[max_value] = pow(fact[max_value], mod - 2, mod)
        for x in range(max_value - 1, 0, -1):
            invs[x] = invs[x + 1] * (x + 1) % mod

        self.fact = fact
        self.invs = invs
        self.mod = mod

    def nCr(self, n, r):
        r = min(n - r, r)
        if r < 0: return 0
        if r == 0: return 1
        if r == 1: return n
        return self.fact[n] * self.invs[r] * self.invs[n - r] % self.mod

    def nHr(self, n, r):
        return self.nCr(n - 1 + r, r)


def main():
    MOD = 10 ** 9 + 7

    r1, c1, r2, c2 = map(int, input().split())

    cl = Calc(max_value=r2 + c2 + 1, mod=MOD)
    nCr = cl.nCr

    ans = 0
    for r in range(r1, r2 + 1):
        ans = (ans + nCr(r + c2 + 1, r + 1) - nCr(r + c1 - 1 + 1, r + 1)) % MOD
    print(ans)


if __name__ == '__main__':
    main()