Catalan Game

A parenthesis string is a string made only of open parentheses ( and close parentheses ). A correct parenthesis string is defined by the following rules.

• The empty string is a correct parenthesis string.
• If $S$ is a correct parenthesis string, then $(S)$ is also a correct parenthesis string.
• If $S$ and $T$ are correct parenthesis strings, then $ST$, the concatenation of the two, is also a correct parenthesis string.

Any parenthesis string that cannot be built by this process is not a correct parenthesis string.

Junhui recently learned that the number of correct parenthesis strings of length $2n$ is the Catalan number $C_n$. Junhui loves correct parenthesis strings, and is upset that they appear too rarely among all parenthesis strings.

Jimin wants to comfort Junhui by showing that there are plenty of correct parenthesis strings. To do so, Jimin wants to win the following game against Junhui.

You are given an incorrect parenthesis string $S$ of length $N$. The two players take turns removing either the first character or the last character of $S$. They continue until there are no characters left to remove. If at some point during the game $S$ becomes a non-empty correct parenthesis string, Jimin wins. If no such moment occurs before the game ends, Junhui wins.

Jimin moves first. Given $N$ and $S$, determine who wins if both players always play optimally.

The first line contains the length $N$ of the parenthesis string.

The second line contains the parenthesis string $S$.

Print the name of the winner when both players play optimally.

Print jimin if Jimin wins, and junhui if Junhui wins.

• $1\leq N\leq 5\times 10^{5}$
• $|S| = N$
• Every character of $S$ is ( or ).
• $S$ is not a correct parenthesis string.