
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 $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a > b > 0$) that passes through the point $P(-\sqrt{3}, \frac{1}{2})$ and has one focus at $(\sqrt{3}, 0)$.
(Ⅰ) Find the equation of the ellipse $E$.
(Ⅱ) If line $l$ passes through point $M(0, \sqrt{2})$ and intersects the ellipse $E$ at points $A$ and $B$, find the maximum value of $|AB|$.

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