
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
In the cartesian plane, consider the curves  $x^2+y^2=r^2$  and  $(xy)^2=1$ . Call  $F_r$  the convex polygon with vertices the points of intersection of these 2 curves. (if they exist)

(a) Find the area of the polygon as a function of  $r$ .
(b) For which values of  $r$  do we have a regular polygon?

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