
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 \( A_{1} A_{2} \) and \( B_{1} B_{2} \) be the diameters of a circle with center \( O \). Determine and geometrically represent the set of all points \( P \) for which the inequalities

\[
A_{1} P > O P, \quad A_{2} P > O P, \quad B_{1} P > O P, \quad B_{2} P > O P
\]

hold simultaneously.

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