#897 OrBiTal (easy)

Statement

Statement language

There are $N$ orbitals in a line, numbered $1$ through $N$ from left to right. Initially, every orbital has force intensity zero.

The particle strength is an integer $K$ satisfying $1\leq K\leq\lfloor(N+1)/2\rfloor$ . A strength- $K$ particle may only be placed at orbital $j$ with . It adds to orbital and, for each integer , adds to both orbitals and . Equivalently, it adds

1,2,\ldots,K-1,K,K-1,\ldots,2,1

to a consecutive segment of length $2K-1$ , in that order. The entire segment must lie within orbitals $1$ through $N$ . The two ends are not connected.

After choosing one strength $K$ , you may place any nonnegative number of particles of that strength. Multiple particles may share an orbital, and their contributions accumulate.

The required final intensities $A_1,A_2,\ldots,A_N$ are given. Find every strength for which all required intensities can be obtained exactly.

Input

The input is given in the following format:

$T$ $\mathrm{case}_1$ $\mathrm{case}_2$

Output

For each test case, let $M$ be the number of valid strengths. Print $M,K_1,K_2,\ldots,K_M$ on one line, separated by spaces, with . If no valid strength exists, print .

Constraints

$1\leq T\leq200\,000$ .
$1\leq N\leq200\,000$ .
$0 \leq_{}$ .

Subtasks

#	Score	Constraints
1	25	$A_i\leq10$ .
2	32	The sum of $N$ over all test cases is at most $1000$ .

Samples

Input

Output