
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 that each vertex of a convex hexagon $ABCDEF$ is to be assigned one of 6 available colors, and the colors of all pairs of adjacent vertices as well as the vertices at the ends of each main diagonal must be different, calculate the number of different ways this hexagon can be colored.

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