
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 $C: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{a^{2} - c^{2}} = 1 \ (a > c > 0)$ have its right focus at $F$. A tangent is drawn at a moving point $P$ on the ellipse and intersects the $x$-axis at point $Q$. Find the equation of the locus of the centroid of triangle $\triangle P Q F$.

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