Reconstruct Array

You are given an integer n (n ≥ 2) representing the number of elements in an unknown array of integers sorted in non-decreasing order. You are also provided with a list of n*(n-1)/2 integers. These integers represent the sums of every unique pair of elements from the unknown array, but they are given in an arbitrary order.

Your task is to reconstruct the original array. If multiple valid arrays exist, return the lexicographically smallest one. If no valid array can produce the given pairwise sums, output Impossible.

Input Format

• The first line contains a single integer, n.
• The second line contains n*(n-1)/2 space-separated integers, representing the pairwise sums.

Output Format

• If a valid array exists, output the n integers (in non-decreasing order) separated by spaces.
• If no valid array exists, output the string Impossible.

Constraints

• 2 ≤ n ≤ 100
• Each pairwise sum is an integer in the range [-10^6, 10^6]
• The original array may contain repeated numbers.

Example

Input:

4
3 4 5 5 6 7

Output:

1 2 3 4

Explanation:
The original array can be reconstructed as [1, 2, 3, 4]. The pairwise sums from this array are:

• 1 + 2 = 3
• 1 + 3 = 4
• 1 + 4 = 5
• 2 + 3 = 5
• 2 + 4 = 6
• 3 + 4 = 7

These match the provided input sums.

Note: Your solution should read from standard input and write to standard output. Ensure your code handles all edge cases based on the given constraints.