Sum of consecutive numbers

Time limit5sMemory limit256 MB

Summary

Write each N up to 1e9 as the shortest sum of two or more consecutive positive integers, or print IMPOSSIBLE.

Level: Medium5 of 10
Topics: Math, Number theory
Solved: No attempts yet

Problem

Some numbers are a sum of two or more consecutive positive integers, and some numbers are not.

Given an integer $N$ , write $N$ as a sum of two or more consecutive positive integers. For example, $10 = 1 + 2 + 3 + 4$ and $24 = 7 + 8 + 9$ .

If several sums exist, print the one with the fewest terms. Once the number of terms is fixed the first term is fixed too, so the shortest sum is unique.

Input

The first line contains the number of test cases $T$ ( $1 \le T \le 100$ ).

Each of the next $T$ lines contains one integer $N$ ( $1 \le N \le 10^9$ ).

Output

Print one line for each test case. If the shortest sum runs from $a$ to $b$ , join all of its terms in this format.

N = a + (a+1) + ... + b

Put one space on each side of the equals sign and of every plus sign. If no such sum exists, print IMPOSSIBLE.

Examples2

Example 1

Input

Expected output

IMPOSSIBLE
10 = 1 + 2 + 3 + 4
24 = 7 + 8 + 9

Example 2

Input

Expected output

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