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: Time Limit Exceeded)

MOD = 10**9+7
r1, c1, r2, c2 = map(int, input().split())
n = 2 * 10 ** 6
from math import factorial

class Facts():
    # 階乗のメモ化
    # 組み合わせ数、順列数の計算を高速に行う
    def __init__(self, max_num=10**5, p=10**9 + 7):
        self.p = p
        self.max_num = max_num
        self.fact = [1] * (self.max_num + 1)
        for i in range(1, self.max_num + 1):
            self.fact[i] = self.fact[i-1] * i
            self.fact[i] %= self.p

    def comb(self, n, k):
        # nCk mod p を求める
        if n < 0 or k < 0 or n < k:
            return 0
        if n == 0 or k == 0:
            return 1
        a = self.fact[n]
        b = self.fact[k]
        c = self.fact[n-k]
        return (a*self.power_func(b, self.p-2) *
                self.power_func(c, self.p-2)) % self.p

    def perm(self, n, k):
        # nPk mod p を求める
        if n < 0 or k < 0 or n < k:
            return 0
        if n == 0 or k == 0:
            return 1
        a = self.fact[n]
        b = self.fact[n-k]
        return (a * self.power_func(b, self.p-2)) % self.p


    def power_func(self, a, b):
        # a^b mod p　を繰り返し二乗法で求める
        if b == 0:
            return 1
        if b % 2 == 0:
            d = self.power_func(a, b//2)
            return d*d % self.p
        if b % 2 == 1:
            return (a*self.power_func(a, b-1)) % self.p

class FLT:
    #フェルマーの小定理
    # a^(-1) = a^(m-2) mod p
    def __init__(self, mod=10**9+7):
        self.mod = mod

    def rep_sqr(self, base, k):
        if k == 0:
            return 1
        elif k % 2 == 0:
            return (self.rep_sqr(base, k // 2) ** 2) % self.mod
        else:
            return (self.rep_sqr(base, k - 1) * base) % self.mod

    def inv(self, a):
        # 逆元を取る
        return self.rep_sqr(a, self.mod-2)

fact = Facts(n)

def f(x, y):
    return fact.comb(x+y, y)

def g(x, y):
    s = 0
    for i in range(y+1):
        s += f(x, i+1)
        s %= MOD
    return s

print((g(r2, c2) - g(r2, c1-1) - g(r1 - 1, c2) + g(r1 - 1, c1 - 1)) % MOD)