Ice Rink Game

AtCoder

IDagc020_b

Time2000ms

Memory256MB

Difficulty—

An adult game master and $N$ children are playing a game on an ice rink. The game consists of $K$ rounds. In the $i$\-th round, the game master announces: * Form groups consisting of $A_i$ children each! Then the children who are still in the game form as many groups of $A_i$ children as possible. One child may belong to at most one group. Those who are left without a group leave the game. The others proceed to the next round. Note that it's possible that nobody leaves the game in some round. In the end, after the $K$\-th round, there are exactly two children left, and they are declared the winners. You have heard the values of $A_1$, $A_2$, ..., $A_K$. You don't know $N$, but you want to estimate it. Find the smallest and the largest possible number of children in the game before the start, or determine that no valid values of $N$ exist. ## Constraints * $1 \leq K \leq 10^5$ * $2 \leq A_i \leq 10^9$ * All input values are integers. ## Input Input is given from Standard Input in the following format: $K$ $A_1$ $A_2$ $...$ $A_K$ [samples]

Samples

Input #1

4
3 4 3 2

Output #1

6 8

For example, if the game starts with $6$ children, then it proceeds as follows:

*   In the first round, $6$ children form $2$ groups of $3$ children, and nobody leaves the game.
*   In the second round, $6$ children form $1$ group of $4$ children, and $2$ children leave the game.
*   In the third round, $4$ children form $1$ group of $3$ children, and $1$ child leaves the game.
*   In the fourth round, $3$ children form $1$ group of $2$ children, and $1$ child leaves the game.

The last $2$ children are declared the winners.

Input #2

5
3 4 100 3 2

Output #2

\-1

This situation is impossible. In particular, if the game starts with less than $100$ children, everyone leaves after the third round.

Input #3

10
2 2 2 2 2 2 2 2 2 2

Output #3

2 3

API Response (JSON)

{
  "problem": {
    "name": "Ice Rink Game",
    "description": {
      "content": "An adult game master and $N$ children are playing a game on an ice rink. The game consists of $K$ rounds. In the $i$\\-th round, the game master announces: *   Form groups consisting of $A_i$ children",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc020_b"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "An adult game master and $N$ children are playing a game on an ice rink. The game consists of $K$ rounds. In the $i$\\-th round, the game master announces:\n\n*   Form groups consisting of $A_i$ children...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments