
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
Points \( A, B, C \) in the plane satisfy \( \overline{AB} = 2002 \) and \( \overline{AC} = 9999 \). The circles with diameters \( AB \) and \( AC \) intersect at \( A \) and \( D \). If \( \overline{AD} = 37 \), what is the shortest distance from point \( A \) to line \( BC \)?

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