
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 combination for a certain lock is set by assigning either the number 0 or 1 to each vertex of a regular $n$-sided polygon $A_{1} A_{2} \cdots A_{n}$ and simultaneously coloring each vertex either red or blue, such that any two adjacent vertices must share at least one same number or color. How many different possible combinations are there for this lock?

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