Root Nim Game

Time limit1sMemory limit512 MB

Summary

Given piles of stones, a move replaces a pile of x with any y in [x^(1/4), sqrt(x)] with y < x; determine the winner under optimal play.

Level: Medium7 of 10
Topics: Game theory, Math, Prefix sum
Solved: No attempts yet

Problem

Koosaga and Cubelover are going to play a Root Nim game. The game uses $N$ piles of stones, and pile $i$ has $A_i$ stones. The two players take turns. On each turn, a player must do the following.

Choose one pile. Let $x$ be the number of stones in that pile.
Choose an integer $y$ that satisfies $0 \le y < x$ and $x^{1/4} \le y \le x^{1/2}$ , then change the number of stones in the chosen pile to $y$ .

A player who cannot make a move loses the game.

Given $N$ and the number of stones in each pile, write a program that determines who wins. Both players play optimally. Koosaga moves first.

Input

The first line contains the number of piles $N$ ( $1 \le N \le 100000$ ). The second line contains $A_i$ ( $1 \le A_i \le 1000000000000$ ), the number of stones in each pile.

Output

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

Examples3

Example 1
Input
```
1
1
```
Expected output
```
cubelover
```
Example 2
Input
```
2
1 2
```
Expected output
```
cubelover
```
Example 3
Input
```
10
1 2 3 4 5 6 7 8 9 10
```
Expected output
```
koosaga
```