
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
An ellipse competition II question. Given a line $e$ passing through the center of a circle, draw tangents from an external point $P$ on $e$ to the circle and find the angle bisectors of these tangents and $e$. What is the locus of the feet of the perpendiculars dropped from the center of the circle to these angle bisectors as point $P$ moves along the portion of $e$ outside the circle?

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