
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 the ellipse $E$: $\frac{x^2}{18} + \frac{y^2}{9} = 1$ ($a > b > 0$) with its right focus $F$ at $(3, 0)$. A line passing through point $F$ intersects $E$ at points $A$ and $B$. If the midpoint of segment $AB$ has coordinates $(1, -1)$, find the equation of $E$.

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