
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
Let \( n \) be a given positive integer. Consider the set of points in the plane Cartesian coordinate system \( xOy \)
\[ T = \{(x, y) \mid |x| = |y| \leq n, \, x \text{ and } y \in \mathbb{Z}\} . \]

For two points \( P \) and \( Q \) in \( T \), \( P \) and \( Q \) are called "adjacent" if and only if \( |PQ| = \sqrt{2} \) or \( PQ \) is parallel to one of the two coordinate axes. Each point in \( T \) is colored with one of three colors: red, blue, or green, such that any two adjacent points are colored differently. Find the number of such coloring ways.

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