It can be solve using fast matrix exponentiation (modified version, based on XOR operation). Simulate XOR rounds one-by-one using variables, and you'll see that there is pattern of matrix which is move. Start from vector B = [1, 1, 0, ..., 1]. Then do the following process: next[i] = XOR (prev[j] & B[(i + j) % n]) k times. Then multiply (the same operation) result by the array A.
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