
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
In triangle \(ABC\), it is known that \(\angle BAC = \alpha\), \(\angle ABC = \beta\), \(BC = a\), and \(AD\) is the altitude. On the side \(AB\), a point \(P\) is taken such that \(\frac{AP}{PB} = \frac{1}{2}\). A circle is drawn through point \(P\), tangent to side \(BC\) at point \(D\). Find the radius of this circle.

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