Two coins of each kind (2)

Given a natural number $x$ and $n$ different coin values $c_1 \ldots c_n$ , compute in how many ways it is possible to achieve change $x$ by using each value at most twice. Here, two coins with the same value are considered different.

For example, if $x = 4$ and the available values are $1$ and $2$ , then there are three ways to achieve it: $1 + 1' + 2$ , $1 + 1' + 2'$ , and also $2 + 2'$ .

Input

Input consists of several cases. Every case begins with $x$ and $n$ , followed by $c_1 \ldots c_n$ . Assume $1 \le n \le 1000$ , $1 \le c_i \le x \le 1000$ , and that all $c_i$ are different.

Output

For every case, print the number of ways to exactly achieve change $x$ by using each value at most twice. Since the result can be huge, make the computations modulo $10^8 + 7$ .

Problem information

Author: Unknown
Translator: Albert Atserias

Generation: 2026-01-25T11:08:48.164Z