
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
A clock has an hour, minute, and second hand, all of length  $1$ . Let  $T$  be the triangle formed by the ends of these hands. A time of day is chosen uniformly at random. What is the expected value of the area of  $T$ ?

*Proposed by Dylan Toh*

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