
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
From a point $P$ on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, a perpendicular line is drawn to the $x$-axis, and the foot of the perpendicular is exactly the left focus $F_1$. Also, point $A$ is the intersection of the ellipse with the positive $x$-axis, and point $B$ is the intersection of the ellipse with the positive $y$-axis, and $AB \parallel OP$. Given that $|F_1A| = \sqrt{10} + \sqrt{5}$, find the equation of the ellipse.

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