Buttons
Contest: Codeforces Round 893 (Div. 2)
Difficulty rating: 800
Tags: games, greedy, math

Anna and Katie ended up in a secret laboratory.

There are $$$a+b+c$$$ buttons in the laboratory. It turned out that $$$a$$$ buttons can only be pressed by Anna, $$$b$$$ buttons can only be pressed by Katie, and $$$c$$$ buttons can be pressed by either of them. Anna and Katie decided to play a game, taking turns pressing these buttons. Anna makes the first turn. Each button can be pressed at most once, so at some point, one of the girls will not be able to make her turn.

The girl who cannot press a button loses. Determine who will win if both girls play optimally.

Input format:
The first line contains a single integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases.

Each test case consists of three integers $$$a$$$, $$$b$$$, and $$$c$$$ ($$$1 \le a, b, c \le 10^9$$$) — the number of buttons that can only be pressed by Anna, the number of buttons that can only be pressed by Katie, and the number of buttons that can be pressed by either of them, respectively.

Output format:
For each test case, output First if Anna wins, or Second if Katie wins.

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

Note:
For the simplicity of the explanation, we will numerate the buttons by the numbers from $$$1$$$ to $$$a+b+c$$$: the first $$$a$$$ buttons can only be pressed by Anna, the next $$$b$$$ buttons can only be pressed by Katie, and the last $$$c$$$ buttons can be pressed by either of them.

In the first test case, Anna can press the $$$3$$$-rd button on the first turn. Then Katie will press the $$$2$$$-nd button (since it is the only possible turn for her). Then Anna will press the $$$1$$$-st button. Katie won't have a button to press, so Anna will win.

In the second test case, Anna can press the first nine buttons in some order on her turns. No matter what buttons Katie will press, all the buttons from the $$$10$$$-th to the $$$15$$$-th will be pressed after $$$12$$$ turns. On the $$$13$$$-th turn, Anna will press one of the first nine buttons and Katie will not have a button to press on her turn. Thus, Anna will win.

In the third test case, the game can proceed as follows:

- On the $$$1$$$-st turn Anna presses the $$$5$$$-th button.
- On the $$$2$$$-st turn Katie presses the $$$4$$$-th button.
- On the $$$3$$$-st turn Anna presses the $$$6$$$-th button.
- On the $$$4$$$-st turn Katie presses the $$$3$$$-th button.
- On the $$$5$$$-st turn Anna presses the $$$1$$$-th button.
- On the $$$6$$$-st turn Katie presses the $$$2$$$-th button.
- Anna cannot make the turn, so Katie wins.

It can be shown that Katie can win no matter what moves Anna takes.

Example input:
5
1 1 1
9 3 3
1 2 3
6 6 9
2 2 8

Example output:
First
First
Second
First
Second

Editorial:
On each turn, the current player gets rid of one of the buttons available to them. At the same time, if you press the "common" button, the enemy will not be able to press it as well. Since each player wants to leave their opponent without buttons to press before they run out of those themselves, they will click on the "common" buttons as long as there is at least one available.
The number of the player who clicks on their (not the general) button first depends on $$$c$$$. It can be noticed that this player will win if and only if they have strictly more buttons than their opponent.