Description

Given the sums of each row and each column of an $n \times 3$ non-negative integer matrix, count how many matrices satisfy these conditions. Output the answer modulo $10^{17}$.

Input Format

The first line contains four positive integers $N$, $c_1$, $c_2$, $c_3$, representing the number of rows and the sums of the three columns. The second line contains $N$ positive integers, representing the sum of the three numbers in each row. The sum of each row and each column does not exceed $125$.

Output Format

Output one number: the number of matrices satisfying the conditions modulo $10^{17}$.

3 2 3 4
1 2 6

17

Hint

Translated by ChatGPT 5