
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 \( ABC \) be a triangle with \( AB = 20 \), \( BC = 10 \), and \( CA = 15 \). Let \( I \) be the incenter of \( ABC \), and let \( BI \) meet \( AC \) at \( E \) and \( CI \) meet \( AB \) at \( F \). Suppose that the circumcircles of \( BIF \) and \( CIE \) meet at a point \( D \) different from \( I \). Find the length of the tangent from \( A \) to the circumcircle of \( DEF \).

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