
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 points $A$ and $B$ on the ellipse $\frac{x^2}{a^2} + \frac{25y^2}{9a^2} = 1$, and $F_2$ is the right focus of the ellipse. If $|AF_2| + |BF_2| = \frac{8}{5}a$, and the distance from the midpoint of $AB$ to the left directrix of the ellipse is $\frac{3}{2}$, find the equation of the ellipse.

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