
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 \(ABC\) is changing in such a way that vertex \(A\) of the right angle of the triangle remains in the same position, while vertices \(B\) and \(C\) slide along fixed circles \(S_{1}\) and \(S_{2}\) that are externally tangent at point \(A\). Find the locus of the feet \(D\) of the altitudes \(AD\) of triangles \(ABC\).

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