
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]


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Given the ellipse \( C: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \) where \( a > b \geqslant 0 \), with an eccentricity of \( \frac{4}{5} \) and the distance between the directrices being \( \frac{25}{2} \),
1. Find the values of \( a \) and \( b \).
2. Given the point \( A \) with coordinates \((6,0)\) and point \( B \) moving on the ellipse \( C \), where \( \triangle ABP \) is an isosceles right triangle with \( A \) as the right-angle vertex and \( A, B, P \) arranged in a clockwise direction, find the equation of the locus of point \( P \).

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