F. Fallen Towerstime limit per test3 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard output Pizano built an array $$$a$$$ of $$$n$$$ towers, each consisting of $$$a_i \ge 0$$$ blocks.Pizano can knock down a tower so that the next $$$a_i$$$ towers grow by $$$1$$$. In other words, he can take the element $$$a_i$$$, increase the next $$$a_i$$$ elements by one, and then set $$$a_i$$$ to $$$0$$$. The blocks that fall outside the array of towers disappear. If Pizano knocks down a tower with $$$0$$$ blocks, nothing happens.Pizano wants to knock down all $$$n$$$ towers in any order, each exactly once. That is, for each $$$i$$$ from $$$1$$$ to $$$n$$$, he will knock down the tower at position $$$i$$$ exactly once.Moreover, the resulting array of tower heights must be non-decreasing. This means that after he knocks down all $$$n$$$ towers, for any $$$i < j$$$, the tower at position $$$i$$$ must not be taller than the tower at position $$$j$$$.You are required to output the maximum $$$\text{MEX}$$$ of the resulting array of tower heights.The $$$\text{MEX}$$$ of an array is the smallest non-negative integer that is not present in the array.InputEach test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10^4$$$). The description of the test cases follows. The first line of each test case contains an integer $$$n$$$ ($$$1 \leq n \leq 10^5$$$) — the number of towers.The second line of each test case contains $$$n$$$ integers — the initial heights of the towers $$$a_1, \ldots, a_n$$$ ($$$0 \leq a_i \leq 10^9$$$).It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$10^5$$$. OutputFor each test case, output a single integer — the maximum $$$\text{MEX}$$$ of the final array.ExampleInput821 242 1 0 0105 9 3 7 1 5 1 5 4 3101 1 1 1 1 1 1 1 1 1103 2 1 0 3 2 1 0 3 255 2 0 5 51100000000074 0 1 0 2 7 7Output2
3
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NoteExplanation for the first test case. Explanation for the second test case. Note that all towers were knocked down exactly once, and the final array of heights is non-decreasing.