CF1832E Combinatorics Problem

Recall that the binomial coefficient $ \binom{x}{y} $ is calculated as follows ( $ x $ and $ y $ are non-negative integers): - if $ x < y $ , then $ \binom{x}{y} = 0 $ ; - otherwise, $ \binom{x}{y} = \frac{x!}{y! \cdot (x-y)!} $ . You are given an array $ a_1, a_2, \dots, a_n $ and an integer $ k $ . You have to calculate a new array $ b_1, b_2, \dots, b_n $ , where - $ b_1 = (\binom{1}{k} \cdot a_1) \bmod 998244353 $ ; - $ b_2 = (\binom{2}{k} \cdot a_1 + \binom{1}{k} \cdot a_2) \bmod 998244353 $ ; - $ b_3 = (\binom{3}{k} \cdot a_1 + \binom{2}{k} \cdot a_2 + \binom{1}{k} \cdot a_3) \bmod 998244353 $ , and so on. Formally, $ b_i = (\sum\limits_{j=1}^{i} \binom{i - j + 1}{k} \cdot a_j) \bmod 998244353 $ . Note that the array is given in a modified way, and you have to output it in a modified way as well.

The only line of the input contains six integers $ n $ , $ a_1 $ , $ x $ , $ y $ , $ m $ and $ k $ ( $ 1 \le n \le 10^7 $ ; $ 0 \le a_1, x, y < m $ ; $ 2 \le m \le 998244353 $ ; $ 1 \le k \le 5 $ ). The array $ [a_1, a_2, \dots, a_n] $ is generated as follows: - $ a_1 $ is given in the input; - for $ 2 \le i \le n $ , $ a_i = (a_{i-1} \cdot x + y) \bmod m $ .

Since outputting up to $ 10^7 $ integers might be too slow, you have to do the following: Let $ c_i = b_i \cdot i $ (without taking modulo $ 998244353 $ after the multiplication). Print the integer $ c_1 \oplus c_2 \oplus \dots \oplus c_n $ , where $ \oplus $ denotes the bitwise XOR operator.