
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 right-angled coordinate system $(xOy)$, a conic section $C$ with the origin as the pole and the positive half of the $x$-axis as the polar axis has the polar coordinate equation $({ρ}^{2}= \frac{12}{3+{\sin }^{2}θ} )$. Point $A$ is $(0,- \sqrt{3})$ and $F_{1}$, $F_{2}$ are the left and right foci of the conic section $C$. A line passes through point $F_{1}$ and is parallel to line $AF_{2}$.

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

(2) If the line intersects conic section $C$ at points $M$ and $N$, find $|F_{1}M|⋅|F_{1}N|$.

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