
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
For \(i=0,1, \ldots, 5\), let \(l_{i}\) be the ray on the Cartesian plane starting at the origin, at an angle \(\theta=i \frac{\pi}{3}\) counterclockwise from the positive \(x\)-axis. For each \(i\), point \(P_{i}\) is chosen uniformly at random from the intersection of \(l_{i}\) with the unit disk. Consider the convex hull of the points \(P_{i}\), which will (with probability 1) be a convex polygon with \(n\) vertices for some \(n\). What is the expected value of \(n\)?

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