A16412 | Training Before the Olympiad

Masha and Olya have an important team olympiad coming up soon. In honor of this, Masha, for warm-up, suggested playing a game with Olya:

There is an array $a$ of size $n$ . Masha goes first, and the players take turns. Each move is described by the following sequence of actions:

$\bullet$ If the size of the array is $1$ , the game ends.

$\bullet$ The player who is currently playing chooses two different indices $i$ , $j$ ( $1 \le i, j \le |a|$ ), and performs the following operation — removes $a_i$ and $a_j$ from the array and adds to the array a number equal to $\lfloor \frac{a_i + a_j}{2} \rfloor \cdot 2$ . In other words, first divides the sum of the numbers $a_i$ , $a_j$ by $2$ rounding down, and then multiplies the result by $2$ .

Masha aims to maximize the final number, while Olya aims to minimize it.

Masha and Olya decided to play on each non-empty prefix of the initial array $a$ , and asked for your help.

For each $k = 1, 2, \ldots, n$ , answer the following question. Let only the first $k$ elements of the array $a$ be present in the game, with indices $1, 2, \ldots, k$ respectively. What number will remain at the end with optimal play by both players?

The first line contains an integer $t$ ( $1 \leq t \leq 10^4$ ) — the number of test cases.

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

The second line contains $n$ integers $a_1,a_2, \ldots,a_n$ ( $1 \leq a_i \leq 10^9$ ) — the array on which Masha and Olya play.

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

For each test case, output $n$ integers. The $k$ -th of these numbers should be equal to the number that will remain at the end with optimal play by both players, on the array consisting of the first $k$ elements of the array $a$ .