One Square in a Triangle

AtCoder

IDarc167_e

Time2000ms

Memory256MB

Difficulty—

A triangle $ABC$ on the $xy$\-plane is said to be a good triangle when it satisfies all of the following conditions. * Each of the vertices $A$, $B$, and $C$ is a lattice point whose $x$\- and $y$\-coordinates are between $0$ and $10^{8}$, inclusive. * The triangle $ABC$ (**including** the perimeter and vertices) wholly contains exactly one square of area $1$ whose vertices are all lattice points. You are given a positive integer $S$. Determine if there is a good triangle of area $\frac{S}{2}$, and construct one if it exists. For each input file, you have $T$ test cases to solve. ## Constraints * $1\leq T\leq 10^{5}$ * $1\leq S\leq 10^{8}$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $T$ $\text{case}_{1}$ $\text{case}_{2}$ $\vdots$ $\text{case}_{T}$ Each case is given in the following format: $S$ [samples]

Samples

Input #1

Output #1

No
Yes
1 1 1 3 3 3
Yes
5 1 7 8 4 5

![image](https://img.atcoder.jp/arc167/d6986726412312ca9a6e022bc8e722ce.png)
In the figure, the left and right triangles correspond to the second and third test cases, respectively.

API Response (JSON)

{
  "problem": {
    "name": "One Square in a Triangle",
    "description": {
      "content": "A triangle $ABC$ on the $xy$\\-plane is said to be a good triangle when it satisfies all of the following conditions. *   Each of the vertices $A$, $B$, and $C$ is a lattice point whose $x$\\- and $y$\\",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc167_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "A triangle $ABC$ on the $xy$\\-plane is said to be a good triangle when it satisfies all of the following conditions.\n\n*   Each of the vertices $A$, $B$, and $C$ is a lattice point whose $x$\\- and $y$\\...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments