
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 triangle \( A B C \) with sides \( A B = \sqrt{17} \), \( B C = 5 \), and \( A C = 4 \). Point \( D \) is taken on the side \( A C \) such that \( B D \) is the altitude of triangle \( A B C \). Find the radius of the circle passing through points \( A \) and \( D \) and tangent at point \( D \) to the circumcircle of triangle \( B C D \).

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