
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  $C$  be the unit circle  $x^{2}+y^{2}=1 .$  A point  $p$  is chosen randomly on the circumference  $C$  and another point  $q$  is chosen randomly from the interior of  $C$  (these points are chosen independently and uniformly over their domains). Let  $R$  be the rectangle with sides parallel to the  $x$  and  $y$ -axes with diagonal  $p q .$  What is the probability that no point of  $R$  lies outside of  $C ?$ 

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