
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 parametric equation of line $l$ as $\begin{cases} x=m+\frac{\sqrt{2}}{2}t \\ y=\frac{\sqrt{2}}{2}t \end{cases} (t \text{ is the parameter})$, establish a polar coordinate system with the coordinate origin as the pole and the positive semi-axis of $x$ as the polar axis. The polar equation of curve $C$ is $\rho^{2}\cos ^{2}\theta+3\rho^{2}\sin ^{2}\theta=12$, and the left focus $F$ of the curve $C$ is on line $l$.

(1) If line $l$ intersects curve $C$ at points $A$ and $B$, find the value of $|FA| \cdot |FB|$;

(2) Find the maximum value of the perimeter of the inscribed rectangle of curve $C$.

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