
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 $E$: $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a > b > 0)$ passing through point $P(-\sqrt{3},\frac{1}{2})$, and one focus of the ellipse $E$ is $(\sqrt{3},0)$.

(I) Find the equation of the ellipse $E$;

(II) If line $l$ passes through point $M(0,\sqrt{2})$ and intersects with ellipse $E$ at points $A$ and $B$, find the maximum value of $|AB|$.

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