
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 triangle is inscribed in a given ellipse such that two vertices of the triangle are at the endpoints of one of the symmetry axes, and the third vertex moves along the ellipse. What is the locus of the orthocenter of the triangle?

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