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

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

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

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