
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 $E: \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 \; (a > b > 0)$ whose right focus is $F(3,0)$, and a line passing through point $F$ intersects the ellipse $E$ at points $A$ and $B$. If the midpoint of $AB$ is $(1,-1)$, then the equation of the ellipse $E$ is ______.

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