Tree Restoring

AtCoder

IDagc005_c

Time2000ms

Memory256MB

Difficulty—

Aoki loves numerical sequences and trees. One day, Takahashi gave him an integer sequence of length $N$, $a_1, a_2, ..., a_N$, which made him want to construct a tree. Aoki wants to construct a tree with $N$ vertices numbered $1$ through $N$, such that for each $i = 1,2,...,N$, the distance between vertex $i$ and the farthest vertex from it is $a_i$, assuming that the length of each edge is $1$. Determine whether such a tree exists. ## Constraints * $2 ≦ N ≦ 100$ * $1 ≦ a_i ≦ N-1$ ## Input The input is given from Standard Input in the following format: $N$ $a_1$ $a_2$ $...$ $a_N$ [samples]

Samples

Input #1

5
3 2 2 3 3

Output #1

Possible

![image](https://atcoder.jp/img/agc005/cda0380bb5cd1b9502cfceaf2526d91e.png)
The diagram above shows an example of a tree that satisfies the conditions. The red arrows show paths from each vertex to the farthest vertex from it.

Input #2

3
1 1 2

Output #2

Impossible

Input #3

10
1 2 2 2 2 2 2 2 2 2

Output #3

Possible

Input #4

10
1 1 2 2 2 2 2 2 2 2

Output #4

Impossible

Input #5

6
1 1 1 1 1 5

Output #5

Impossible

Input #6

5
4 3 2 3 4

Output #6

Possible

API Response (JSON)

{
  "problem": {
    "name": "Tree Restoring",
    "description": {
      "content": "Aoki loves numerical sequences and trees. One day, Takahashi gave him an integer sequence of length $N$, $a_1, a_2, ..., a_N$, which made him want to construct a tree. Aoki wants to construct a tree w",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc005_c"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Aoki loves numerical sequences and trees.\nOne day, Takahashi gave him an integer sequence of length $N$, $a_1, a_2, ..., a_N$, which made him want to construct a tree.\nAoki wants to construct a tree w...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments