
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 either red or blue with equal probability. What is the probability that there will be at least one triangle whose vertices are among the vertices of the hexagon and all of whose sides are the same color?
A) $\frac{31}{32}$
B) $\frac{253}{256}$
C) $\frac{1021}{1024}$
D) $\frac{4093}{4096}$
E) $\frac{65535}{65536}$

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