Score : $600$ points

问题描述

输出满足以下两个条件的整数序列长度为 $N$ 的序列 $A = (A_1, A_2, \ldots, A_N)$ 的个数，结果对 $998244353$ 取模。

• $0 \leq A_1 \leq A_2 \leq \cdots \leq A_N \leq M$
• $A_1 \oplus A_2 \oplus \cdots \oplus A_N = X$

此处，$\oplus$ 表示按位异或运算。

什么是按位异或运算？

非负整数 $A$ 和 $B$ 的按位异或运算（记作 $A \oplus B$）定义如下：

• 当 $A \oplus B$ 以二进制表示时，第 $k$ 位最低位（$k \geq 0$）在以下情况下为 $1$：$A$ 和 $B$ 的第 $k$ 位最低位中恰好有一个是 $1$；否则为 $0$。

例如，$3 \oplus 5 = 6$（二进制表示：$011 \oplus 101 = 110$）。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

Print the number of integer sequences of length $N$, $A = (A_1, A_2, \ldots, A_N)$, that satisfy both of the following conditions, modulo $998244353$.

• $0 \leq A_1 \leq A_2 \leq \cdots \leq A_N \leq M$
• $A_1 \oplus A_2 \oplus \cdots \oplus A_N = X$

Here, $\oplus$ denotes bitwise XOR.

What is bitwise XOR?

The bitwise XOR of non-negative integers $A$ and $B$, $A \oplus B$, is defined as follows.

• When $A \oplus B$ is written in binary, the $k$-th lowest bit ($k \geq 0$) is $1$ if exactly one of the $k$-th lowest bits of $A$ and $B$ is $1$, and $0$ otherwise.

For instance, $3 \oplus 5 = 6$ (in binary: $011 \oplus 101 = 110$).

Constraints

• $1 \leq N \leq 200$
• $0 \leq M \lt 2^{30}$
• $0 \leq X \lt 2^{30}$
• All values in the input are integers.

Input

The input is given from Standard Input in the following format:

$N$ $M$ $X$

Output

Print the answer.

Sample Input 1

3 3 2

Sample Output 1

5

Here are the five sequences of length $N$ that satisfy both conditions in the statement: $(0, 0, 2), (0, 1, 3), (1, 1, 2), (2, 2, 2), (2, 3, 3)$.

Sample Input 2

200 900606388 317329110

Sample Output 2

788002104

update @ 2024/3/10 12:03:19