
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
Po picks 100 points \( P_{1}, P_{2}, \ldots, P_{100} \) on a circle independently and uniformly at random. He then draws the line segments connecting \( P_{1} P_{2}, P_{2} P_{3}, \ldots, P_{100} P_{1} \). When all of the line segments are drawn, the circle is divided into a number of regions. Find the expected number of regions that have all sides bounded by straight lines.

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