Increasing Prefix XOR

AtCoder

IDarc141_b

Time2000ms

Memory256MB

Difficulty—

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: $N$ $M$ [samples]

Samples

Input #1

2 4

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 $(A_1,\ A_2)=(1,\ 2),\ (1,\ 4),\ (2,\ 4),\ (3,\ 4)$.

Input #2

4 4

Output #2

Input #3

10 123456789

Output #3

205695670

API Response (JSON)

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    "name": "Increasing Prefix XOR",
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      "content": "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",
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      "statement_type": "Markdown",
      "content": "You are given positive integers $N$ and $M$.\nFind the number of sequences $A=(A_1,\\ A_2,\\ \\dots,\\ A_N)$ of $N$ positive integers that satisfy the following conditions, modulo $998244353$.\n\n*   $1 \\leq...",
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