
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
A right triangle \( ABC \) changes in such a way that the vertex \( A \) of the right angle remains fixed, while the vertices \( B \) and \( C \) slide along fixed circles \( S_1 \) and \( S_2 \), respectively, which touch externally at the point \( A \). Find the locus of the feet of the altitudes \( AD \) of the triangles \( ABC \).

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