
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  $\omega$  be the unit circle centered at the origin of  $R^2$ . Determine the largest possible value for the radius of the circle inscribed to the triangle  $OAP$  where  $ P$  lies the circle and  $A$  is the projection of  $P$  on the axis  $OX$ .

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