23132번 - Nimber Sequence 다국어

Given are Nimbers $a_1, a_2, \ldots, a_{K-1}$, $b_1, b_2, \ldots, b_5$, and $c_1, c_2, \ldots, c_5$.

Let $\displaystyle a_n = \left(\bigoplus_{i = 1}^{5} a_{n - i} \otimes b_i \right) \oplus \left(\bigoplus_{i = 1}^{5} a_{n - K + i} \otimes c_i \right)$ for all $n \ge K$.

Find the value of $a_m$.

The first line of input contains two positive integers $K$ and $m$ ($6 \le K \le 10^5$, $1 \le m \le 10^{18}$).

The second line contains $K - 1$ non-negative integers $a_1, a_2, \ldots, a_{K - 1}$ ($0 \le a_i < 2^{32}$).

The third line contains five non-negative integers $b_1, b_2, \ldots, b_5$ ($0 \le b_i < 2^{32}$).

The fourth line contains five non-negative integers $c_1, c_2, \ldots, c_5$ ($0 \le c_i < 2^{32}$).

Output a single line with a single integer: the value of $a_m$.

Nimbers are non-negative integers associated with Nim games. For the purposes of this problem, two operations are necessary:

• "$\large\oplus$" is the Nim sum: $\displaystyle a \oplus b = \mathrm{mex}\left( \{a' \oplus b \mid 0 \le a' < a\} \cup \{a \oplus b' \mid 0 \le b' < b\} \right)$,
• "$\large\otimes$" is the Nim product: $\displaystyle a \otimes b = \mathrm{mex}\left( \{(a' \otimes b) \oplus (a \otimes b') \oplus (a' \otimes b') \mid 0 \le a' < a, \, 0 \le b' < b \} \right)$.

Here, $\mathrm{mex}(S)$ represents the smallest non-negative integer $d \not\in S$.