2019湖南省赛 4 Buttons（推公式）

4 Buttons
Bobo lives in an infinite chessboard. Initially he locates at $(0, 0)$. There are $4$ buttons.

When the first button is pressed, Bobo moves right for at most $a$ cells.
When the second button is pressed, Bobo moves up for at most $b$ cells.
When the third button is pressed, Bobo moves left for at most $c$ cells.
When the fourth button is pressed, Bobo moves down for at most $d$ cells.
Find the number of cells Bobo can reach modulo $(10^9+7)$, if he presses the buttons for no more than $n$ times.

Input
The input consists of several test cases and is terminated by end-of-file.

Each test case contains five integers $n$, $a$, $b$, $c$ and $d$.

$1 \leq n, a, b, c, d \leq 10^9$
The number of test cases does not exceed $10^5$.
Output
For each test case, print an integer which denotes the result.

Note
For the first test case, Bobo can reach the following $11$ cells: $(-3, 0)$, $(-2, 0)$, $(-1, 0)$, $(0, -4)$, $(0, -3)$, $(0, -2)$, $(0, -1)$, $(0, 0)$, $(0, 1)$, $(0, 2)$, $(1, 0)$.

思路：
画出图可以发现
会出现底层n - 1个方格，顶层1个方格，一共n-1层，一共四部分，面积分别为 $a*b 和 b*c 和 c*d 和 d*a$，
加上一个 “十字架” ，长度为 t2 = 1 + (a + b + c + d) * n。

#include <cstdio>
#include <cstring>
#include <algorithm>
#include <vector>
#include <list>

using namespace std;

typedef long long ll;
const ll mod = 1e9 + 7;

int main()
{
    ll n,a,b,c,d;
    while(~scanf("%lld%lld%lld%lld%lld",&n,&a,&b,&c,&d))
    {
        ll t1 = ((n - 1) + 1) * (n - 1) / 2 % mod * (a * b % mod + b * c % mod + c * d % mod + d * a % mod) % mod;
        ll t2 = (1 + (a + b + c + d) * n) % mod;;
        printf("%lld\n",(t1 + t2) % mod);
    }
    return 0;
}