
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 a circle centered at \( O \) with some geometric constructions: \( PA \) is tangent to the circle at point \( A \), \( PC \) intersects the circle at points \( B \) and \( C \), and \( PO \) intersects the circle at point \( D \). Line \( AE \) is perpendicular to \( PO \) at point \( E \). Line \( BE \) is extended to intersect the circle at point \( F \), and segments \( OC \), \( OF \), \( AD \), and \( AF \) are connected. Given \(\angle BCO = 30^\circ \) and \(\angle BFO = 20^\circ \), find the measure of \(\angle DAF \).

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