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 LuaJIT (2.0.4) (judge verdict: Time Limit Exceeded)

local mod = 1000000007
local mfl = math.floor

local function bmul(x, y)
  local x0, y0 = x % 31623, y % 31623
  local x1, y1 = mfl(x / 31623), mfl(y / 31623)
  return (x1 * y1 * 14122 + (x1 * y0 + x0 * y1) * 31623 + x0 * y0) % mod
end

local function badd(x, y)
  return (x + y) % mod
end

local function bsub(x, y)
  return x < y and x - y + mod or x - y
end
local function modpow(src, pow)
  local res = 1
  while 0 < pow do
    if pow % 2 == 1 then
      res = bmul(res, src)
      pow = pow - 1
    end
    src = bmul(src, src)
    pow = mfl(pow / 2)
  end
  return res
end

local function modinv(src)
  return modpow(src, mod - 2)
end

local r1, c1 = io.read("*n", "*n")
local r2, c2 = io.read("*n", "*n")

local function getSum(r, c)
  local sum = 0
  local cur = 1
  for ir = 0, r do
    cur = bmul(cur, ir + c + 1)
    cur = bmul(cur, modinv(ir + 1))
    sum = badd(sum, cur)
  end
  return sum
end
local z = getSum(r2, c2)
z = bsub(z, getSum(r2, c1 - 1))
z = bsub(z, getSum(r1 - 1, c2))
z = badd(z, getSum(r1 - 1, c1 - 1))
print(z)