
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 that the center of ellipse $C$ is at the origin $O$, its foci are on the $x$-axis, and its eccentricity is $\frac{\sqrt{6}}{3}$. Point $B$ is the lower endpoint of the minor axis of the ellipse, and the distance from $B$ to one focus of the ellipse is $\sqrt{3}$.
(I) Find the equation of ellipse $C$;
(II) Let line $l$ pass through point $P(0, \frac{3}{2})$ and intersect ellipse $C$ at points $M$ and $N$, with $|BM|=|BN|$. Find the equation of line $l$.

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