Squarks

A physicist named Byteasar is studying squarks, a newly discovered constituent of matter. Squarks are exotic particles that never exist alone but always in pairs, and a squark of one kind only ever pairs with a squark of a different kind.

After years of study he has established that there are $n$ different kinds of squarks. Each kind has a unique mass that is a positive integer multiple of some fixed unit. He has also measured the total mass of each of the $\frac{n(n-1)}{2}$ possible pairs of two different kinds. According to the standard model, the mass of a pair equals the sum of the masses of the two squarks forming it.

Now he wants to determine the individual mass of each kind of squark. Write a program that reconstructs every mass configuration that is consistent with his measurements.

The first line contains an integer $n$ ($3 \le n \le 300$), the number of different kinds of squarks.

The second line contains the total masses of all $\frac{n(n-1)}{2}$ possible pairs, separated by single spaces. All of them are positive integers, and the mass of each pair does not exceed $10^8$. For every two different kinds of squarks, the mass of the pair they form is given exactly once, in an arbitrary order.

In tests worth 32% of the points it additionally holds that $n \le 20$ and the mass of every pair does not exceed 2000.

On the first line print the number $k$ of possible solutions. It is guaranteed that at least one solution exists for every input, so $k > 0$.

On each of the following $k$ lines print one solution: the $n$ distinct positive integer masses of all kinds, in increasing order, separated by single spaces.

So that the answer is uniquely determined, print the $k$ solutions in lexicographically increasing order when each is compared as a sequence of integers.