P7487 "Stoi2031" Lanting Xu.

> Unrelated to romance, I write the preface and wait for you to return. > With a suspended brush, I compose a unique masterpiece, while waves pile up in layers on that shore. > How to explain the word “love”? No matter how I write, it is not right. > Yet what I lack is your understanding throughout my life. > — "Lanting Xu".

Yue likes complex numbers very much, especially complex numbers of the form $e^{2\pi it}$. She chooses two positive integers $n,k$, and calls $1+e^{\frac{2\pi i x_1 \dots x_k}{n}}$ the **absolute degree** of $(x_1,\dots,x_k)$. The product of the **absolute degrees** of all $(x_1,\dots,x_k)$ satisfying $1 \le x_i \le n$ $(i \in \{1,2,\dots,k\})$ is called the **irrelevance degree** of $(n,k)$. Now she wants you to help her compute, for each $t \in \{1,2,\dots,k\}$, the **irrelevance degree** of $(n,t)$, denoted as $ans \bmod{335544323}$. Since it is too troublesome to report too many numbers, you only need to output the result of XOR-ing all answers together.

One line with two positive integers $n,k$.

One line with one non-negative integer representing the answer.

#### Brief statement of the problem: Given $n,k$, for $1 \le t \le k$ compute $$\prod_{x_1=1}^{n}\prod_{x_2=1}^{n}\dots\prod_{x_t=1}^{n}\left( 1+e^{\frac{2\pi ix_1x_2\dots x_t}{n}} \right) \bmod{335544323}$$ Output the XOR-sum of all $k$ answers. Here, $e^{it}=\cos{t}+i\sin{t}$ holds for all $t \in \mathbb{R}$, and $i$ is the imaginary unit, satisfying $i^2=-1$. #### Sample explanation: For the first sample, when $t=1,2$, the answers are $2,35184372088832$ respectively. After taking modulo, they are $2,201012021$, and the XOR result is $201012023$. For the second sample, when $t=1,2,3$, all answers are $2$, and the XOR result is $2$. Due to space limits, the remaining samples are not explained. #### Constraints: **This problem uses bundled testcases. The limits and scores of each subtask are as follows:** | Subtask No. | $n \le$ | $k \le$ | Special constraint | Score | | :-----------: | :-----------: | :-----------: | :-----------: | :-----------: | | $1$ | $10$ | $1$ | None | $7$ | | $2$ | $1$ | $10$ | None | $7$ | | $3$ | $10$ | $2$ | None | $7$ | | $4$ | $10^{18}$ | $10^5$ | $n$ is even | $7$ | | $5$ | $10$ | $10$ | $n^k \le 730$ | $16$ | | $6$ | $10^9$ | $10^3$ | None | $19$ | | $7$ | $10^{18}$ | $10^5$ | None | $37$ | For $100\%$ of the testdata, $1 \le n \le 10^{18},1 \le k \le 10^5$. Translated by ChatGPT 5