
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 $n$-sided polygon ($n \geqslant 3$) where each vertex is colored such that the endpoints of every edge have different colors, and $m$ ($m \geqslant 3$) different colors are available, the total number of distinct coloring methods is given by:
$$
b_{n}=(m-1)\left[(m-1)^{n-1}+(-1)^{n}\right] .
$$

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