L Partition

AtCoder

IDarc190_b

Time2000ms

Memory256MB

Difficulty—

There is an $N \times N$ grid. Let $(i,j)$ denote the cell at the $i$\-th row from the top and the $j$\-th column from the left. For $K = 1, 2, \ldots, N$, a **level $K$ L-shape** is a set of $2K - 1$ cells that satisfies at least one of the following four conditions: * All cells reachable from a certain cell $(i,j)$ by moving down or right between $0$ and $K-1$ cells, inclusive (where $1 \leq i \leq N-K+1$, $1 \leq j \leq N-K+1$). * All cells reachable from a certain cell $(i,j)$ by moving down or left between $0$ and $K-1$ cells, inclusive (where $1 \leq i \leq N-K+1$, $K \leq j \leq N$). * All cells reachable from a certain cell $(i,j)$ by moving up or right between $0$ and $K-1$ cells, inclusive (where $K \leq i \leq N$, $1 \leq j \leq N-K+1$). * All cells reachable from a certain cell $(i,j)$ by moving up or left between $0$ and $K-1$ cells, inclusive (where $K \leq i \leq N$, $K \leq j \leq N$). You are given a cell $(a,b)$ and $Q$ queries $k_1, \ldots, k_Q$. For each $i$, print the number, modulo $998244353$, of ways to partition the entire grid into exactly one level $1$ L-shape, one level $2$ L-shape, $\ldots$, and one level $N$ L-shape so that cell $(a,b)$ is contained in the level $k_i$ L-shape. ## Constraints * $1 \leq N \leq 10^7$ * $1 \leq a \leq N$ * $1 \leq b \leq N$ * $1 \leq Q \leq \min\lbrace N, 200000 \rbrace$ * $1 \leq k_1 < \cdots < k_Q \leq N$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $a$ $b$ $Q$ $k_1$ $\cdots$ $k_Q$ [samples]

Samples

Input #1

3 1 2
1
2

Output #1

6

The six ways shown in the following figure are the solutions. In the figure, an integer $k$ in a cell means that the cell belongs to the level $k$ L-shape.
![image](https://img.atcoder.jp/arc190/d853bd693f0d3848c725803512dc382a.png)

Input #2

5 2 5
3
1 3 5

Output #2

4
32
128

Input #3

100 50 50
4
1 10 50 100

Output #3

API Response (JSON)

{
  "problem": {
    "name": "L Partition",
    "description": {
      "content": "There is an $N \\times N$ grid. Let $(i,j)$ denote the cell at the $i$\\-th row from the top and the $j$\\-th column from the left. For $K = 1, 2, \\ldots, N$, a **level $K$ L-shape** is a set of $2K - 1$",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc190_b"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There is an $N \\times N$ grid. Let $(i,j)$ denote the cell at the $i$\\-th row from the top and the $j$\\-th column from the left.\nFor $K = 1, 2, \\ldots, N$, a **level $K$ L-shape** is a set of $2K - 1$...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments