
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 $F\_1$ and $F\_2$ are the left and right foci of the ellipse $E$: $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1 (a > b > 0)$, $M$ is a point on the ellipse $E$, the midpoint of the line segment $F\_1M$ is $N$, and the perimeter of $\triangle NF\_1O$ (where $O$ is the coordinate origin) is $3$. A line passing through the right focus $F\_2$ and perpendicular to the $x$-axis intersects the ellipse $E$ in the first quadrant at point $C$, with $|{{F}\_{2}}C|=\frac{3}{2}$.

(1) Find the standard equation of ellipse $C$;

(2) A line $l$ passing through $F\_1$ intersects the ellipse $E$ at points $A$ and $B$. $P(-5,0)$, and a parallelogram $PAQB$ is formed with $PA$ and $PB$ as adjacent sides. Find the range of the area of the parallelogram $PAQB$.

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