Ex - make 1

AtCoder

IDabc278_h

Time3000ms

Memory256MB

Difficulty—

A sequence $S$ of non-negative integers is said to be a **good sequence** if: * there exists a non-empty (not necessarily contiguous) subsequence $T$ of $S$ such that the bitwise XOR of all elements in $T$ is $1$. There are an empty sequence $A$, and $2^B$ cards with each of the integers between $0$ and $2^B-1$ written on them. You repeat the following operation until $A$ becomes a good sequence: * You freely choose a card and append the integer written on it to the tail of $A$. Then, you eat the card. (Once eaten, the card cannot be chosen anymore.) How many sequences of length $N$ can be the final $A$ after the operations? Find the count modulo $998244353$. What is bitwise 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 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 * $1 \leq N \leq 2 \times 10^5$ * $1 \leq B \leq 10^7$ * $N \leq 2^B$ * $N$ and $B$ are integers. ## Input The input is given from Standard Input in the following format: $N$ $B$ [samples]

Samples

Input #1

2 2

Output #1

5

The following five sequences of length $2$ can be the final $A$ after the operations.

*   $(0, 1)$
*   $(2, 1)$
*   $(2, 3)$
*   $(3, 1)$
*   $(3, 2)$

Input #2

2022 1119

Output #2

293184537

Input #3

200000 10000000

Output #3

383948354

API Response (JSON)

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    "name": "Ex - make 1",
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      "content": "A sequence $S$ of non-negative integers is said to be a **good sequence** if: *   there exists a non-empty (not necessarily contiguous) subsequence $T$ of $S$ such that the bitwise XOR of all element",
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    "platform": "AtCoder",
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      "memory_limit": 262144
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    "sign": "abc278_h"
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      "statement_type": "Markdown",
      "content": "A sequence $S$ of non-negative integers is said to be a **good sequence** if:\n\n*   there exists a non-empty (not necessarily contiguous) subsequence $T$ of $S$ such that the bitwise XOR of all element...",
      "is_translate": false,
      "language": "English"
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}

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