
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
The equation $\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$ represents an ellipse with foci $F_1$ and $F_2$, and vertex $A$. Given that the area of $\triangle AF_1F_2$ is $\sqrt{3}$ and $\angle F_1AF_2=4\angle AF_1F_2$, find the equation of the ellipse.

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