
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
In the rectangular coordinate system $(xOy)$, a point $P(1, \frac{3}{2})$ lies on the ellipse $(C)$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$. The sum of the distances from $P$ to the two foci of the ellipse $(C)$ is $4$.
(1) Find the equation of the ellipse $(C)$;
(2) If points $M$, $N$ are on the ellipse $(C)$ and quadrilateral $POMN$ is a parallelogram, determine the coordinates of points $M$ and $N$.

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