
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
A certain password lock is set such that each vertex of a regular $n$-gon $A_1 A_2 \cdots A_n$ is assigned one of two numbers, 0 or 1, and also painted in one of two colors, red or blue. The assignment must ensure that any two adjacent vertices share at least the same number or the same color. How many different password configurations are there for such a lock?

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