
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
Given that line \( l \) intersects sphere \( O \) at exactly one point \( P \), and two half-planes \( \alpha \) and \( \beta \) emanating from line \( l \) intersect the sphere \( O \) to form two circular cross-sections with radii \( 1 \) and \( \sqrt{3} \), respectively, and the dihedral angle \( \alpha-l-\beta \) has a plane angle of \( \frac{5 \pi}{6} \), determine the radius of sphere \( O \).

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