
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
From point \(A\), lying on a circle, chords \(AB\), \(AC\), and tangent \(AD\) are drawn. The angle between the chords is \(\frac{\pi}{6}\), and the angle between the chord \(AC\) and the tangent \(AD\), which does not contain the chord \(AB\), is \(\frac{5\pi}{12}\). Compute the integer part of the radius of the circle if the area of the triangle \(ABC\) is 32.

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