Minimal Operations for Generating a Set

Given a closed interval \([A,B]\) and an initially empty set, you are allowed to perform the following operation: choose a real number \(x\) and add all numbers in the sequence \(0, x, 2x, 3x, \dots\) that lie in \([A,B]\) to the set (duplicates are ignored). You are provided with the final set \(S\) of numbers (which is guaranteed to be achievable by some sequence of operations). Your task is to find a sequence of operations (i.e. a set of values \(x\)) that, when applied starting from the empty set, produces exactly \(S\); moreover, the number of operations must be minimized.

A key observation is that every operation produces an arithmetic progression starting at \(0\) (so \(0 \in S\) always) of the form:

[ P(x)={kx\mid k\ge0,, kx\in[A,B]} ]

For each chosen \(x\), it is required that \(P(x) \subseteq S\). The goal is to select as few such \(x\) values as possible so that the union of the corresponding progressions equals \(S\) exactly.

inputFormat

The input consists of multiple lines. The first line contains two real numbers \(A\) and \(B\) denoting the interval \([A,B]\). The second line contains an integer \(N\), the number of elements in the final set \(S\). The third line contains \(N\) space‐separated real numbers representing the elements of \(S\) (not necessarily sorted). It is guaranteed that \(0 \in S\) and that \(S\) is exactly the union of several arithmetic progressions starting at 0 as described above.

outputFormat

Output the minimal number \(k\) of operations on the first line. On the second line, output \(k\) space‐separated real numbers representing the chosen values \(x\) (in any order or sorted in increasing order) that yield exactly the set \(S\) when applying the operation described.

sample

0 10
6
0 2 4 6 8 10

1
2