
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
The central angle of sector $AOB$ is $2\theta$, where $\theta\in(0, \frac{\pi}{2})$, and its radius is $r$. An inscribed circle $O_{1}$ within sector $AOB$ that is tangent to circle $O_{1}$, and tangent to $OA$, $OB$ is made. What value of $\sin \theta$ makes the area of circle $O_{2}$ the largest? What is the maximum value?

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