
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
The key to the cipher known as the "turning grille" consists of a stencil made from a square sheet of checkered paper with size $n \times n$ (where $n$ is even). Some cells on the stencil are cut out. One side of the stencil is marked.

When this stencil is placed over a clean sheet of paper in any of four possible orientations (with the marked side up, right, down, or left), its cutouts completely cover the entire area of the square, with each cell being under a cutout exactly once. The letters of the message, which has a length of $n^{2}$, are sequentially written into the cutouts of the stencil, starting with the stencil placed on the clean sheet with the marked side up. After filling all the cutouts with the message letters, the stencil is repositioned, and so on. After removing the stencil, an encrypted message remains on the paper.

Find the number of different keys for an arbitrary even number $n$.

Present the answer in LaTex format: \boxed{Your answer}
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