
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 conic section $C$ has the polar coordinate equation $({{\rho }^{2}}=\dfrac{12}{3+{{\sin }^{2}}\theta })$ with the coordinate origin as the pole and the positive half of the $x$-axis as the polar axis. Point $A(0,-\sqrt{3})$, and $F\_1$, $F\_2$ are the left and right foci of the conic section $C$, respectively. The line $l$ passes through point $F\_1$ and is parallel to the line $AF\_2$.

(1) Find the rectangular coordinate equation of conic section $C$ and the parametric equation of line $l$;

(2) If line $l$ intersects conic section $C$ at points $M$ and $N$, find $|F\_1M| \cdot |F\_1N|$.

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