
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
We color all vertexs of a convex polygon with  $10$  vertexs by  $2$  colors: red and blue  $($ each vertex is colored by  $1$  color $).$ How many ways to color all the vertexs such that there are no  $2$  adjacent vertex that are both colored red?

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