
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-angled 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 $S_{2}$, which are externally tangent at point $A$. Find the locus of the feet of the altitudes $D$ from $A$ to $BC$ in the triangles $ABC$.

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