
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
Consider a regular hexagon where each of its $6$ sides and the $9$ diagonals are colored randomly and independently either red or blue, each color with the same probability. What is the probability that there exists at least one triangle, formed by three of the hexagon’s vertices, in which all sides are of the same color?
A) $\frac{253}{256}$
B) $\frac{1001}{1024}$
C) $\frac{815}{819}$
D) $\frac{1048575}{1048576}$
E) 1

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