
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 \(AOB\) be a 60-degree angle. For any point \(P\) in the interior of \(\angle AOB\), let \(A'\) and \(B'\) be the feet of the perpendiculars from \(P\) to \(AO\) and \(BO\) respectively. Denote by \(r\) and \(s\) the distances \(OP\) and \(A'B'\). Find all possible pairs of real numbers \((r, s)\).

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