Sequential Nim
Contest: Codeforces Round 658 (Div. 2)
Difficulty rating: 1100
Tags: dp, games

There are $$$n$$$ piles of stones, where the $$$i$$$-th pile has $$$a_i$$$ stones. Two people play a game, where they take alternating turns removing stones.

In a move, a player may remove a positive number of stones from the first non-empty pile (the pile with the minimal index, that has at least one stone). The first player who cannot make a move (because all piles are empty) loses the game. If both players play optimally, determine the winner of the game.

Input format:
The first line contains a single integer $$$t$$$ ($$$1\le t\le 1000$$$)  — the number of test cases. Next $$$2t$$$ lines contain descriptions of test cases.

The first line of each test case contains a single integer $$$n$$$ ($$$1\le n\le 10^5$$$)  — the number of piles.

The second line of each test case contains $$$n$$$ integers $$$a_1,\ldots,a_n$$$ ($$$1\le a_i\le 10^9$$$)  — $$$a_i$$$ is equal to the number of stones in the $$$i$$$-th pile.

It is guaranteed that the sum of $$$n$$$ for all test cases does not exceed $$$10^5$$$.

Output format:
For each test case, if the player who makes the first move will win, output "First". Otherwise, output "Second".

Constraints:
Time limit: 1.0 s
Memory limit: 256.0 MB

Note:
In the first test case, the first player will win the game. His winning strategy is:

1. The first player should take the stones from the first pile. He will take $$$1$$$ stone. The numbers of stones in piles will be $$$[1, 5, 4]$$$.
2. The second player should take the stones from the first pile. He will take $$$1$$$ stone because he can't take any other number of stones. The numbers of stones in piles will be $$$[0, 5, 4]$$$.
3. The first player should take the stones from the second pile because the first pile is empty. He will take $$$4$$$ stones. The numbers of stones in piles will be $$$[0, 1, 4]$$$.
4. The second player should take the stones from the second pile because the first pile is empty. He will take $$$1$$$ stone because he can't take any other number of stones. The numbers of stones in piles will be $$$[0, 0, 4]$$$.
5. The first player should take the stones from the third pile because the first and second piles are empty. He will take $$$4$$$ stones. The numbers of stones in piles will be $$$[0, 0, 0]$$$.
6. The second player will lose the game because all piles will be empty.

Example input:
7
3
2 5 4
8
1 1 1 1 1 1 1 1
6
1 2 3 4 5 6
6
1 1 2 1 2 2
1
1000000000
5
1 2 2 1 1
3
1 1 1

Example output:
First
Second
Second
First
First
Second
First

Editorial:
Suppose $$$a_1>1$$$. If removing the entire first pile is winning, player 1 will do that. Otherwise, player 1 can leave exactly one stone in the first pile, forcing player 2 to remove it, leaving player 1 in the winning position. Otherwise, if $$$a_1=1$$$, then it is forced to remove the first pile.
So, whichever player gets the first pile with more than one stone wins. That is, let $$$k$$$ be the maximum number such that $$$a_1=\cdots=a_k=1$$$. If $$$k$$$ is even, the first player will win. Otherwise, the second player will win. The only exception is when all piles have exactly $$$1$$$ stone. In that case, the first player wins when $$$k$$$ is odd.
Complexity is $$$O(n)$$$.