
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
An \((n-1) \times (n-1)\) grid is divided into \((n-1)^2\) cells, and each of the \(n^2\) vertices of these cells is colored either red or blue. Determine the number of different ways to color these vertices such that each cell has exactly two red vertices. Two colorings are considered different if at least one vertex is colored differently.

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