2065: [USACO 2025 US Open Contest, Bronze] Problem 2. More Cow Photos

The cows are in a particularly mischievous mood today! All Farmer John wants to do is take a photograph of the cows standing in a line, but they keep moving right before he has a chance to snap the picture.

Specifically, each of FJ's N cows (1 ≤ N ≤ 10⁵) has an integer height from 1 to N. FJ wants to take a picture of the cows standing in line in a very specific ordering. If the cows have heights h₁, …, h_K when lined up from left to right, he wants the cow heights to have the following three properties:

• He wants the cow heights to increase and then decrease. Formally, there must exist an integer i such that h₁≤ ⋯ ≤ h_i ≥ ⋯ ≥ h_K.
• He does not want any cow standing next to another cow with exactly the same height. Formally, h_i ≠ h_i+1 for all 1 ≤ i < K.
• He wants the picture to be symmetric. Formally, if i + j = K + 1, then h_i = h_j.

FJ wants the picture to contain as many cows as possible. Specifically, FJ can remove some cows and rearrange the remaining ones. Compute the maximum number of cows FJ can have in the picture satisfying his constraints.

The first line of input contains a single integer T (1 ≤ T ≤ 10⁵) denoting the number of test cases. T test cases follow.

The first line of every test case contains a single integer N. The second line of every test case contains N integers, the heights of the N cows available. The cow heights will be between 1 and N.

It is guaranteed the sum of N over all test cases will not exceed 10⁶.

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