Background

$$Walking among the world,$$ $$may not be angels.$$ $$Those who bring disaster,$$ $$may not be demons.$$ $$A demon's tears are treasured by angels,$$ $$and an angel's purity is guarded by demons.$$ $$Perhaps,$$ $$this is the best ending.$$

Problem Description

Some preliminary definitions:

• Elements in a multiset must be non-negative integers.

• The size of a multiset is the number of elements in the multiset.

• For a multiset of size $x$, let its elements be $a_1,a_2\dots a_x$. Then the value of this multiset is $a_1\oplus a_2\oplus \dots \oplus a_x$, that is, the xor sum of all elements in the multiset.

Now given $n,m$.

How many different multisets $S$ of size $n$ satisfy:

$$\max\limits_{T \subseteq S,T\ne \emptyset}{Q_T}=m$$

where $Q_T$ is the value of multiset $T$.

Note: Due to the nature of multisets, multisets with the same numbers but different orders are considered the same multiset. For example, $\left \{ 1,2,3 \right \}$ and $\left \{ 3,2,1 \right \}$ are the same multiset.

Output the number of different multisets modulo $998244353$.

It can be proven that the number of such multisets is finite.

Input Format

The first line contains two integers, representing $n$ and $m$.

Output Format

One line with one integer, the answer modulo $998244353$.

3 5

13

12 7

48643

Hint

Sample Explanation

In sample 1, the $13$ valid multisets are:

$$(0,0,5)$$, $$(0,1,4)$$, $$(0,1,5)$$, $$(0,4,5)$$, $$(0,5,5)$$, $$(1,1,4)$$, $$(1,1,5)$$, $$(1,4,4)$$, $$(1,4,5)$$, $$(1,5,5)$$, $$(4,4,5)$$, $$(4,5,5)$$, $$(5,5,5)$$. #### Constraints | subtask ID | $n$ | $m$ | Special Property | Score | | :----------: | :----------: | :----------: | :----------: | :-----: | | $0$ | $\le 10$ | $\le 10$ | $-$ | $10$ | | $1$ | $\le 10^5$ | $<2^{60}$ | $A$ | $20$ | | $2$ | $\le 2000$ | $\le 2000$ | $-$ | $10$ | | $3$ | $\le 10^5$ | $<2^{60}$ | $-$ | $60$ | Special property $A$: $\operatorname{popcount}(m)\le 5\ $, where $\operatorname{popcount}(m)$ is the number of $1$ bits in the binary representation of $m$. For all testdata, it is guaranteed that $1\le n\le 10^5$, $0\le m<2^{60}$. **Special reminder: This problem uses bundled subtask testing. You can get the score of a subtask only if you pass all test points in that subtask.** Translated by ChatGPT 5$$