CF2199I Strange Process

You are given three arrays: - array $ a $ , initially consisting of $ n $ ones; - array $ b $ , initially consisting of $ m $ zeros; - array $ c $ , consisting of $ m $ integers from $ 1 $ to $ 50 $ . We perform the following process on these arrays. The process consists of $ m $ stages. During the $ i $ -th stage, the following occurs: 1. first, for each element of the array $ a $ , one of three actions is independently chosen: it is multiplied by some prime number, divided by some prime number (it cannot be divided by a number that does not divide the element evenly), or remains unchanged. You don't have to choose the same action or prime number for each element of $ a $ ; 2. then you can either skip this step or choose any $ k $ such that $ a_k = c_i $ , and assign $ b_i = k $ . An array of $ m $ numbers from $ 0 $ to $ n $ is called achievable if it can be obtained as a result of this process as the array $ b $ . Your task is to count the number of achievable arrays.

The first line contains two integers $ n $ and $ m $ ( $ 1 \le n, m \le 10^4 $ ). The second line contains $ m $ integers $ c_1, c_2, \dots, c_m $ ( $ 1 \le c_i \le 50 $ ).

Print one integer — the number of achievable arrays, taken modulo $ 998244353 $ .