Ex - Odd Steps

AtCoder

IDabc258_h

Time3000ms

Memory256MB

Difficulty—

Find the number, modulo $998244353$, of sequences $X$ that satisfy all of the following conditions. * Every term in $X$ is a positive **odd number**. * The sum of the terms in $X$ is $S$. * The prefix sums of $X$ contain none of $A_1, \dots, A_N$. Formally, if we define $Y_i = X_1 + \dots + X_i$ for each $i$, then $Y_i \neq A_j$ holds for all integers $i$ and $j$ such that $1 \leq i \leq |X|$ and $1 \leq j \leq N$. ## Constraints * $1 \leq N \leq 10^5$ * $1 \leq A_1 \lt A_2 \lt \dots \lt A_N \lt S \leq 10^{18}$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $S$ $A_1$ $\ldots$ $A_N$ [samples]

Samples

Input #1

3 7
2 4 5

Output #1

3

The following three sequences satisfy the conditions.

*   $(1, 5, 1)$
*   $(3, 3, 1)$
*   $(7)$

Input #2

5 60
10 20 30 40 50

Output #2

37634180

Input #3

10 1000000000000000000
1 2 4 8 16 32 64 128 256 512

Output #3

75326268

API Response (JSON)

{
  "problem": {
    "name": "Ex - Odd Steps",
    "description": {
      "content": "Find the number, modulo $998244353$, of sequences $X$ that satisfy all of the following conditions. *   Every term in $X$ is a positive **odd number**. *   The sum of the terms in $X$ is $S$. *   The",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 3000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc258_h"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Find the number, modulo $998244353$, of sequences $X$ that satisfy all of the following conditions.\n\n*   Every term in $X$ is a positive **odd number**.\n*   The sum of the terms in $X$ is $S$.\n*   The...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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