The game of Nim (3)

This is another problem about the game of Nim, which is thoroughly explained in problem problem://problemsjutge.org:problems/upc/2004-semi1/5.pbm. But here, we may have a huge number of sets, each with a huge number of marbles. Furthermore, now we assume that the player to make the last move loses, instead of winning.

Input

Input consists of several cases. Every case begins with the number of sets $n$ , followed by the number of marbles of each set, all between 0 and $10^9$ . Assume $0 \le n \le 10^5$ . At least one set has one or more marbles.

Output

For every case, tell if it is a winning or a losing configuration.

Hint

You should use a mathematical trick to solve this problem.

Problem information

Author: Salvador Roura

Generation: 2026-01-25T11:16:01.289Z