
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 $\frac{x^{2}}{4}+ \frac{y^{2}}{3}=1$, a moving line $l$ intersects the ellipse at points $B$ and $C$ ($B$ is in the first quadrant).
(1) If the coordinates of point $B$ are $(1, \frac{3}{2})$, find the maximum area of $\triangle OBC$.
(2) Let $B(x_{1},y_{1})$ and $C(x_{2},y_{2})$, and $3y_{1}+y_{2}=0$, find the equation of line $l$ when the area of $\triangle OBC$ is maximum.

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