Self Checkout

AtCoder

IDttpc2024_1_b

Time2000ms

Memory256MB

Difficulty—

You are given a sequence $S$ of length $N$, consisting of integers $1$, $2$, and $3$. Determine the number of sequences $A$ consisting of integers $1$ and $2$ such that the sequence $T$ obtained after performing the following procedure matches $S$. Output the answer modulo $998244353$. It can be proven that the number of such sequences $A$ is finite. > Start with an empty sequence $T$. Given a sequence $A$ consisting of integers $1$ and $2$, perform the following process to obtain the sequence $T$: > > 1. Set a variable $C = 0$. > 2. If $A$ contains at least one $1$, remove the first occurrence of $1$ from $A$ and add $1$ to $C$. > 3. If $A$ is not empty, remove the first element $x$ of $A$ and add $x$ to $C$. > 4. Append $C$ to the end of $T$. > 5. If $A$ is empty, terminate the process. Otherwise, return to step 1. ## Constraints * All input values are integers. * $1 \le N \le 10^6$ * $1 \le S_i \le 3$ ## Input The input is given from Standard Input in the following format: $N$ $S_1$ $S_2$ $\dots$ $S_N$ [samples]

Samples

Input #1

2
3 2

Output #1

5

There are $5$ possible sequences $A$ that result in $T = (3, 2)$: $A = (1, 2, 2), (2, 1, 2), (2, 2, 1), (2, 1, 1, 1), (1, 2, 1, 1)$.
For example, for $A = (2, 1, 1, 1)$, the process proceeds as follows:

*   Remove the first occurrence of $1$ in $A$, which is $A_2 = 1$. Now $A = (2, 1, 1)$ and $C = 1$.
*   Remove the first element of $A$, which is $A_1 = 2$. Now $A = (1, 1)$ and $C = 3$.
*   Append $C$ to the end of $T$. Now $T = (3)$.
*   Remove the first occurrence of $1$ in $A$, which is $A_1 = 1$. Now $A = (1)$ and $C = 1$.
*   Remove the first element of $A$, which is $A_1 = 1$. Now $A = ()$ and $C = 2$.
*   Append $C$ to the end of $T$. Now $T = (3, 2)$.

Input #2

6
3 2 2 3 2 1

Output #2

Input #3

5
3 2 1 3 2

Output #3

0

Note that there may be cases where no sequence $A$ produces $S$, in which case the answer is $0$.

API Response (JSON)

{
  "problem": {
    "name": "Self Checkout",
    "description": {
      "content": "You are given a sequence $S$ of length $N$, consisting of integers $1$, $2$, and $3$. Determine the number of sequences $A$ consisting of integers $1$ and $2$ such that the sequence $T$ obtained after",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "ttpc2024_1_b"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given a sequence $S$ of length $N$, consisting of integers $1$, $2$, and $3$. Determine the number of sequences $A$ consisting of integers $1$ and $2$ such that the sequence $T$ obtained after...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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