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Armed Bandit (Hard)

Limits: 1s, 512 MB · Custom Checker

You are in a casino, and you are anxious to play on the new one-armed bandit – a slot machine they just installed.

The slot machine has n wheels in a row. The i-th wheel from the left contains the integers from 1 to ki, inclusive. When you pull the handle, the wheels will spin for a while. When they stop, each wheel will show you one of its integers. The integers are randomly chosen. All random choices are uniform and mutually independent.

You are given the number of wheels n and the size of every wheel ki.

Suppose we ignore the spaces between the wheels and read the sequence of numbers as one long string of digits. Find the sequence of numbers that produces the lexicographically smallest such string.

When comparing two different strings S and T, find the smallest index i at which they differ. The one with a smaller character at that index is lexicographically smaller than the other. If there is no such character (i.e., one of the strings is a prefix of the other), the shorter string is the lexicographically smaller one.

For example:

22 is lexicographically smaller than 220 because 22 is a prefix of 220
123 is lexicographically smaller than 14 because 2 is less than 4.

Input

The first line of the input file contains an integer t (t ≤ 100) specifying the number of test cases. Each test case is preceded by a blank line.

Each test case begins with a line consisting of a single integers n. The following line contains n space-separated integers k1, … , kn (1 ≤ n ≤ 1000, 1 ≤ ki ≤ 100).

Output

For each test case, output a single line with n integers separated by single spaces: the numbers as they appear on the individual wheels of the machine.

Samples

InputOutput
2

3
6 6 6

2
10 2
1 1 1
10 1

This IPSC problem is licensed under the CC BY-SA 3.0 license. The problem statement has been modified slightly to adhere to Toph’s style guide. The custom checker has been modified to be compatible with Toph.

You can find the original problem in the IPSC archive.

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