
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 fixed circumcircle is given around an acute-angled triangle \(ABC\) with vertex \(C\). The orthocenter \(H\) moves along a circle centered at point \(C\). Find the locus of the midpoints of the segments connecting the feet of the altitudes drawn from vertices \(A\) and \(B\).

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