
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
Let  $n, k \geq 3$  be integers, and let  $S$  be a circle. Let  $n$  blue points and  $k$  red points be
chosen uniformly and independently at random on the circle  $S$ . Denote by  $F$  the intersection of the
convex hull of the red points and the convex hull of the blue points. Let  $m$  be the number of vertices
of the convex polygon  $F$  (in particular,  $m=0$  when  $F$  is empty). Find the expected value of  $m$ .


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