1867C · Salyg1n and the MEX Game

Problem: This is an interactive problem!

salyg1n gave Alice a set $S$ of $n$ distinct integers $s_1, s_2, \ldots, s_n$ ($0 \leq s_i \leq 10^9$). Alice decided to play a game with this set against Bob. The rules of the game are as follows:

• Players take turns, with Alice going first.
• In one move, Alice adds one number $x$ ($0 \leq x \leq 10^9$) to the set $S$. The set $S$ must not contain the number $x$ at the time of the move.
• In one move, Bob removes one number $y$ from the set $S$. The set $S$ must contain the number $y$ at the time of the move. Additionally, the number $y$ must be strictly smaller than the last number added by Alice.
• The game ends when Bob cannot make a move or after $2 \cdot n + 1$ moves (in which case Alice's move will be the last one).
• The result of the game is $\operatorname{MEX}\dagger(S)$ ($S$ at the end of the game).
• Alice aims to maximize the result, while Bob aims to minimize it.

Let $R$ be the result when both players play optimally. In this problem, you play as Alice against the jury program playing as Bob. Your task is to implement a strategy for Alice such that the result of the game is always at least $R$.

$\dagger$ $\operatorname{MEX}$ of a set of integers $c_1, c_2, \ldots, c_k$ is defined as the smallest non-negative integer $x$ which does not occur in the set $c$. For example, $\operatorname{MEX}(\{0, 1, 2, 4\})$ $=$ $3$.

Input Format: The first line contains an integer $t$ ($1 \leq t \leq 10^5$) - the number of test cases.

Note: In the first test case, the set $S$ changed as follows:

{$1, 2, 3, 5, 7$} $\to$ {$1, 2, 3, 5, 7, 8$} $\to$ {$1, 2, 3, 5, 8$} $\to$ {$1, 2, 3, 5, 8, 57$} $\to$ {$1, 2, 3, 8, 57$} $\to$ {$0, 1, 2, 3, 8, 57$}. In the end of the game, $\operatorname{MEX}(S) = 4$, $R = 4$.

In the second test case, the set $S$ changed as follows:

{$0, 1, 2$} $\to$ {$0, 1, 2, 3$} $\to$ {$1, 2, 3$} $\to$ {$0, 1, 2, 3$}. In the end of the game, $\operatorname{MEX}(S) = 4$, $R = 4$.

In the third test case, the set $S$ changed as follows:

{$5, 7, 57$} $\to$ {$0, 5, 7, 57$}. In the end of the game, $\operatorname{MEX}(S) = 1$, $R = 1$.