
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 a cipher, called a "turn grille", is a stencil made from a square sheet of graph paper of size $n \times n$ (where $n$ is even). Some cells are cut out. One side of the stencil is marked. When this stencil is laid over a blank sheet of paper in the four possible ways (marked side up, right, down, left), its cutouts completely cover the entire area of the square, with each cell falling 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, first placed on a blank sheet of paper with the marked side up. After filling all the cutouts of the stencil with letters of the message, the stencil is placed in the next position, and so on. After removing the stencil, an encrypted message is left on the sheet of paper.

Find the number of different keys for any even number $n$.

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