
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 Cartesian coordinate system, the parametric equation of line $l$ is given by $\begin{cases} x=m+\frac{\sqrt{2}}{2}t \\ y=\frac{\sqrt{2}}{2}t \end{cases}$ ($t$ is the parameter). Establish a polar coordinate system with the origin as the pole and the positive semi-axis of $x$ as the polar axis. The polar equation of curve $C$ is $\rho^2\cos^2\theta+3\rho^2\sin^2\theta=12$. Point $F$ has polar coordinates $(2\sqrt{2},\pi)$, and $F$ lies 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 rectangle inscribed in curve $C$.

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