
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
Given an integer \( n \geq 2 \). For the set of lattice points in the plane coordinate system
\[ A = \{ (a, b) \mid a, b \in \{1, 2, \cdots, n\} \} \]
each point is colored with one of three colors: red, yellow, or blue. This coloring must satisfy the condition: for any \( a, b \in \{1, 2, \cdots, n-1\} \), if the points \((a, b)\) and \((a+1, b)\) are the same color, then the points \((a, b+1)\) and \((a+1, b+1)\) must also be the same color. Find the total number of different coloring ways.

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