Defeating Bosses

Little X is about to face n bosses. For each boss i, he has already decided on a method b_i (either 1 or 2) to challenge the boss. The rules are as follows:

1. If he chooses method 1 (solo), he defeats boss i on his own and obtains k_i,0 pieces of rare metal. In this case, the rewards for the other theoretical roles are defined by a guarantee:
   \( k_{i,0} = k_{i,1} = k_{i,2} \).
2. If he chooses method 2 (cooperative with Little Y), then ideally Little X should get k_i,1 pieces and Little Y should get k_i,2 pieces of rare metal. However, since the boss's strength is independent of the number of fighters, the system awards both players k_i,0 pieces. In addition, the rewards satisfy the relation: \[ \frac{1}{k_{i,0}} = \frac{1}{k_{i,1}} + \frac{1}{k_{i,2}}. \] It can be shown that if all values are positive integers then for method 2 the solutions \( (k_{i,0}, k_{i,1}, k_{i,2}) \) obey the equation \[ (k_{i,1}-k_{i,0})(k_{i,2}-k_{i,0}) = k_{i,0}^2, \] so that for a given choice of k_i,0 the number of choices for \( (k_{i,1}, k_{i,2}) \) equals the number of positive divisors of \( k_{i,0}^2 \) (denoted by \(\tau(k_{i,0}^2)\)).

Overall, Little X will defeat all bosses and obtain a total of \[ m = \sum_{i=1}^{n} k_{i,0}. \] You are given q queries. In each query a positive integer m is provided. For each query, determine the total number of possible assignments of positive integers \( k_{i,0} \) (and corresponding \( k_{i,1}, k_{i,2} \) as above) for all bosses such that the total pieces sum to m. When boss i uses method 1, there is exactly 1 way (since \( k_{i,0}=k_{i,1}=k_{i,2} \)). When boss i uses method 2, for a chosen k_{i,0} there are \(\tau(k_{i,0}^2)\) valid assignments. Two overall assignments are considered different if at least one \( k_{i,0} \) (or one of the corresponding values) differs.

Since the answer can be very large, output it modulo 998244353.

inputFormat

The first line contains two integers n and q --- the number of bosses and the number of queries.

The second line contains n integers b₁, b₂, ..., b_n, where each b_i is either 1 or 2, indicating the method chosen for boss i.

Then follow q lines. Each of these lines contains a single positive integer m, representing the total amount of rare metal obtained.

outputFormat

For each query, output a single line containing the number of valid assignments modulo 998244353.

sample

2 3
1 2
2
3
4

1
4
7