Ground Squirrels 3

Mog studies the burrows of two ground squirrels. The ground is an $N$-dimensional grid, whose $i$-th axis has length $L_i$. Each cell has integer coordinates $(x_1,x_2,\cdots,x_N)$ with $1 \leq x_i \leq L_i$ ($1 \leq i \leq N$). Two cells are adjacent if they differ by $1$ in exactly one coordinate and agree in every other coordinate.

The first squirrel starts at $A=(a_1,a_2,\cdots,a_N)$, and the second starts at . These starting cells are distinct. Each burrow must satisfy all of the following conditions.

• It contains the starting cell of its squirrel.
• Movement is allowed between every pair of adjacent cells belonging to that burrow.
• Between any two distinct cells of the burrow, there is exactly one path that stays within the burrow and does not visit any cell twice.

A burrow consisting of one cell is allowed. The two burrows must not share a cell.

Determine whether every cell can belong to exactly one of the two burrows. If it is possible, construct one such partition.

The input is given from Standard Input in the following format:

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

For each test case, if no valid partition exists, print NO.

Otherwise, print YES, followed by $V=\prod_{i=1}^{N}L_i$ integers, one for each cell. List cells in lexicographic order of their coordinates. Print for a cell in the first burrow, or for a cell in the second burrow. At the first coordinate where two cells differ, the cell with the smaller value comes first. Thus, the last coordinate varies fastest.

• $1 \leq T \leq 10\,000$.
• $3 \leq N \leq 10$.
• $2 \leq L_i \leq 100$ ().