201411: [AtCoder]ARC141 B - Increasing Prefix XOR

Memory Limit:1024 MB Time Limit:2 S
Judge Style:Text Compare Creator:
Submit:0 Solved:0

Description

Score : $500$ points

Problem Statement

You are given positive integers $N$ and $M$.

Find the number of sequences $A=(A_1,\ A_2,\ \dots,\ A_N)$ of $N$ positive integers that satisfy the following conditions, modulo $998244353$.

  • $1 \leq A_1 < A_2 < \dots < A_N \leq M$.
  • $B_1 < B_2 < \dots < B_N$, where $B_i = A_1 \oplus A_2 \oplus \dots \oplus A_i$.

Here, $\oplus$ denotes bitwise $\mathrm{XOR}$.

What is bitwise $\mathrm{XOR}$?

The bitwise $\mathrm{XOR}$ of non-negative integers $A$ and $B$, $A \oplus B$, is defined as follows:

  • When $A \oplus B$ is written in base two, the digit in the $2^k$'s place ($k \geq 0$) is $1$ if exactly one of $A$ and $B$ is $1$, and $0$ otherwise.
For example, we have $3 \oplus 5 = 6$ (in base two: $011 \oplus 101 = 110$).
Generally, the bitwise $\mathrm{XOR}$ of $k$ non-negative integers $p_1, p_2, p_3, \dots, p_k$ is defined as $(\dots ((p_1 \oplus p_2) \oplus p_3) \oplus \dots \oplus p_k)$. We can prove that this value does not depend on the order of $p_1, p_2, p_3, \dots, p_k$.

Constraints

  • $1 \leq N \leq M < 2^{60}$
  • All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$

Output

Print the answer.


Sample Input 1

2 4

Sample Output 1

5

For example, $(A_1,\ A_2)=(1,\ 3)$ counts, since $A_1 < A_2$ and $B_1 < B_2$ where $B_1=A_1=1,\ B_2=A_1\oplus A_2=2$.

The other pairs that count are $(A_1,\ A_2)=(1,\ 2),\ (1,\ 4),\ (2,\ 4),\ (3,\ 4)$.


Sample Input 2

4 4

Sample Output 2

0

Sample Input 3

10 123456789

Sample Output 3

205695670

Input

题意翻译

给定 $m,n$,希望构造长度为 $m$ 的数列 $a$ 满足条件: - $1\le a_1<a_2<\dots a_m\le n$。 - 令 $b_i=\oplus_{j=1}^i$,那么 $b$ 是单增的。 求方案个数。

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