
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
Each of the $6$ sides and the $9$ diagonals of a regular hexagon are randomly and independently colored red or blue with equal probability. What is the probability that there will be a triangle whose vertices are among the vertices of the hexagon such that all of its sides have the same color?
A) $\frac{399}{4096}$
B) $\frac{105}{128}$
C) $\frac{125}{128}$
D) $\frac{253}{256}$
E) $1$

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