CF1575C Cyclic Sum

Denote a cyclic sequence of size $ n $ as an array $ s $ such that $ s_n $ is adjacent to $ s_1 $ . The segment $ s[r, l] $ where $ l < r $ is the concatenation of $ s[r, n] $ and $ s[1, l] $ . You are given an array $ a $ consisting of $ n $ integers. Define $ b $ as the cyclic sequence obtained from concatenating $ m $ copies of $ a $ . Note that $ b $ has size $ n \cdot m $ . You are given an integer $ k $ where $ k = 1 $ or $ k $ is a prime number. Find the number of different segments in $ b $ where the sum of elements in the segment is divisible by $ k $ . Two segments are considered different if the set of indices of the segments are different. For example, when $ n = 3 $ and $ m = 2 $ , the set of indices for segment $ s[2, 5] $ is $ \{2, 3, 4, 5\} $ , and for segment $ s[5, 2] $ is $ \{5, 6, 1, 2\} $ . In particular, the segments $ s[1, 6], s[2,1], \ldots, s[6, 5] $ are considered as the same segment. Output the answer modulo $ 10^9 + 7 $ .

The first line contains three integers $ n $ , $ m $ , and $ k $ ( $ 1 \leq n, m, k \leq 2 \cdot 10^5 $ , $ k = 1 $ or $ k $ is a prime number). The second line contains $ n $ integers $ a_1, a_2, \ldots, a_n $ ( $ 0 \leq a_i \leq 2 \cdot 10^5 $ ).

Output an integer denoting the number of different segments in $ b $ where the sum of elements in the segment is divisible by $ k $ , modulo $ 10^9 + 7 $ .

In the first example, all valid segments are $ [1,4] $ , $ [2, 3] $ , $ [3, 5] $ , and $ [4, 2] $ . In the second example, one of the valid segments is $ [1, 5] $ .