
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
A unit square \(ABCD\) and a circle \(\Gamma\) have the following property: if \(P\) is a point in the plane not contained in the interior of \(\Gamma\), then \(\min (\angle APB, \angle BPC, \angle CPD, \angle DPA) \leq 60^\circ\). The minimum possible area of \(\Gamma\) can be expressed as \(\frac{a \pi}{b}\) for relatively prime positive integers \(a\) and \(b\). Compute \(100a+b\).

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