
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
P is a point on the interior or boundary of a regular pentagon. Let $r_{1}, r_{2}, r_{3}, r_{4}, r_{5}$ denote the distances from $P$ to the sides of the pentagon, ordered such that $r_{1} \leq r_{2} \leq r_{3} \leq r_{4} \leq r_{5}$. Determine the positions of $P$ where $r_{3}$ is the largest possible, and also those where $r_{3}$ is the smallest possible.

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