
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  $\mathcal{P}$  be a regular  $2022$ -gon with area  $1$ . Find a real number  $c$  such that, if points  $A$  and  $B$  are chosen independently and uniformly at random on the perimeter of  $\mathcal{P}$ , then the probability that  $AB \geq c$  is  $\tfrac{1}{2}$ . 

*Espen Slettnes*

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