Prime Power Game

Time limit1sMemory limit512 MB

Summary

Two players alternately choose a prime power p^k that divides some element and divide every element it divides by p^k; the player who cannot move loses, so print the winner.

Level: Hard8 of 10
Topics: Math, Number theory, Game theory, Bit manipulation
Solved: No attempts yet

Problem

Koosaga and CubeLover are playing the prime power game. The game uses a sequence of length $N$ , $A_1, A_2, \ldots, A_N$ . The two players take turns.

On each turn, the player chooses an integer $p$ and an integer $k$ . Here $p$ must be prime and $k$ must be a positive integer. Also, $p^k$ must divide at least one of the integers in the sequence. Then, every number in the sequence that is divisible by $p^k$ is divided by $p^k$ . The player who cannot choose $p$ and $k$ loses the game.

Write a program that determines which player wins when both players play optimally. Koosaga takes the first turn.

Input

The first line contains $N$ ( $1 \le N \le 100$ ), the size of the sequence. The second line contains $A_1, A_2, \ldots, A_N$ ( $1 \le A_i \le 10^9$ ).

Output

If Koosaga wins, print "koosaga". If CubeLover wins, print "cubelover".

Examples4

Example 1
Input
```
4
1 1 1 1
```
Expected output
```
cubelover
```
Example 2
Input
```
4
1 1 17 17
```
Expected output
```
koosaga
```
Example 3
Input
```
4
1 1 17 289
```
Expected output
```
cubelover
```
Example 4
Input
```
5
1 2 3 4 5
```
Expected output
```
cubelover
```