
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 \( P \) move on the right branch of the hyperbola \( \frac{x^{2}}{16} - \frac{y^{2}}{9} = 1 \) (excluding the two vertices of the hyperbola). Points \( E \) and \( F \) are the left and right foci, respectively, and \( A \) is the excenter of \( \angle P E F \) (the intersection of the internal angle bisector of \( \angle P E F \) and the external angle bisectors of \( \angle P E F \) and \( \angle E P F \)).

Find the equation of the locus of point \( A \).

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