
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
Let the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a>b>0)\), with foci \( F_{1}(-c, 0) \) and \( F_{2}(c, 0) \). Let \( Q \) be a point on the ellipse not at the vertices of the major axis. Draw the perpendicular from the focus \( F_{1} \) (or \( F_{2} \)) to the external bisector of \(\angle F_{1} Q F_{2}\), with the foot of the perpendicular being \( P \). Then the locus of point \( P \) is a circle centered at the origin with a radius \( a \), excluding the points \( (-a, 0) \) and \( (a, 0) \).

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