
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 $(C)$ centered at the origin with the equation $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1$ and one of its foci $F_{1}(0,3)$. Let $M(x,4)$ ($x > 0$) be a point on the ellipse such that the area of $\triangle MOF_{1}$ is $\frac{3}{2}$.

(1) Find the equation of the ellipse $(C)$;
(2) Determine if there exists a line $(l)$ parallel to $OM$ that intersects ellipse $(C)$ at points $A$ and $B$, such that the circle with diameter $AB$ passes through the origin. If it exists, find the equation of the line $(l)$; otherwise, explain why not.

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