CF2237C Duck Surplus

Ja the Ghost is playing with rubber ducks again! There are $ n $ piles of rubber ducks arranged in a row from left to right. Initially, the $ i $ -th pile contains $ a_i $ rubber ducks. While the sequence $ a $ is not sorted in nondecreasing order, Ja must perform the following operation: - Choose two adjacent piles such that the left pile contains more ducks than the right pile. Ja swaps these two piles, and then adds the number of ducks in the new left pile to the new right pile.Formally, choose an index $ i $ such that $ 1\le i \lt n $ and $ a_i \gt a_{i+1} $ . Then replace the adjacent pair $ (a_i,a_{i+1}) $ with $ (a_{i+1},a_i+a_{i+1}) $ . For example, if two adjacent piles contain $ 7 $ and $ 3 $ rubber ducks, then after the operation they contain $ 3 $ and $ 10 $ rubber ducks. Ja may choose any index satisfying the condition above at each step. It can be shown that, regardless of his choices, the process eventually ends with the sequence sorted in nondecreasing order. Ja wants the largest pile at the end of the process to contain as few rubber ducks as possible. Determine the minimum possible value of the largest pile.

Each 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 $ n $ ( $ 1 \le n \le 2 \cdot 10^5 $ ) — the number of piles. The second line of each test case contains $ n $ integers $ a_1, a_2, \ldots, a_n $ ( $ 1\le a_i\le 10^9 $ ) — the number of ducks in each pile. It is guaranteed that the sum of $ n $ over all test cases does not exceed $ 2 \cdot 10^5 $ .

For each test case, output a single integer — the minimum possible value of the largest pile.

In the transformations below, the two underlined numbers are the adjacent pair just obtained by the operation. In the first test case, the sequence is already sorted in nondecreasing order. Therefore, Ja does not perform any operation, and the answer is $5$. In the second test case, Ja has only one possible operation: $$ [7,3]\to[\underline{3},\underline{10}]. $$ The sequence is then sorted, so the answer is $10$. In the third test case, Ja can perform the following operations: $$ [3,2,1]\to[\underline{2},\underline{5},1]\to[2,\underline{1},\underline{6}]\to[\underline{1},\underline{3},6]. $$ The largest pile contains $6$ ducks. If Ja first chooses the last two piles instead, the final largest pile would contain $7$ ducks. Therefore, the answer is $6$. In the fourth test case, Ja cannot choose the first two piles at the beginning, because $2$ is not greater than $2$. One possible process is $$ [2,2,1,3,3]\to[2,\underline{1},\underline{3},3,3]\to[\underline{1},\underline{3},3,3,3]. $$ Thus, the answer is $3$. In the fifth test case, one optimal process is $$ [3,1,4,2]\to[\underline{1},\underline{4},4,2]\to[1,4,\underline{2},\underline{6}]\to[1,\underline{2},\underline{6},6]. $$ Therefore, the answer is $6$.