
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
Given an ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ where $a > b \geq 0$ and its eccentricity is $\frac{4}{5}$, and one of the directrices has the equation $x=\frac{25}{4}$:
1. Find the equation of the ellipse.
2. Given the point $A$ with coordinates $(6, 0)$ and $B$ is a point on the ellipse $C$. Construct an isosceles right triangle $\triangle ABP$ with $A$ as the right-angle vertex and vertices $A, B, P$ in a clockwise direction. Find the equation of the locus of point $P$.

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