AT_fps_24_q サイコロ

There are two dice, called die $ 1 $ and die $ 2 $ . - Die $ 1 $ is an $ N $ -sided die, where each side shows a distinct positive integer $ A_1, A_2, \dots, A_N $ with equal probability. - Die $ 2 $ is an $ M $ -sided die, where each side shows a distinct positive integer $ B_1, B_2, \dots, B_M $ with equal probability. You are also given a positive integer $ K $ . For each $ k = 1, 2, \dots, K $ , solve the following problem: - When rolling die $ 1 $ and die $ 2 $ simultaneously, compute the expected value of $ (\text{sum of the outcomes})^k $ modulo $ 998244353 $ . What does expected value modulo $ 998244353 $ mean? It can be proven that the expected value is always a rational number. Under the constraints of this problem, if the value can be expressed as $ \tfrac{P}{Q} $ with coprime integers $ P $ and $ Q $ , then there exists a unique integer $ R $ such that $ R \times Q \equiv P \pmod{998244353} $ and $ 0 \leq R < 998244353 $ . You are asked to compute this $ R $ .

The input is given from standard input in the following format: > $ N $ $ M $ $ K $ $ A_1 $ $ A_2 $ $ \dots $ $ A_N $ $ B_1 $ $ B_2 $ $ \dots $ $ B_M $

Print $ K $ lines. On the $ i $ -th line, output the answer for $ k = i $ .

### Sample Explanation 1 For example, if die $ 1 $ shows $ 3 $ and die $ 2 $ shows $ 5 $ , then the cube of the sum is $ (3+5)^3 = 512 $ . ### Constraints - $ 1 \leq N \leq 10^5 $ - $ 1 \leq M \leq 10^5 $ - $ 1 \leq K \leq 10^5 $ - $ 1 \leq A_1 < A_2 < \dots < A_N \leq 10^8 $ - $ 1 \leq B_1 < B_2 < \dots < B_M \leq 10^8 $ - All input values are integers