
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 half of the $x$-axis as the polar axis. The polar equation of the ellipse $(C)$ is $ρ^{2}\cos ^{2}θ+3ρ^{2}\sin ^{2}θ=12$. The left focus $(F)$ of the ellipse is located on line $(l)$.
$(1)$ If line $(l)$ intersects ellipse $(C)$ at points $A$ and $B$, find the value of $|FA|⋅|FB|$;
$(2)$ Find the maximum value of the perimeter of the inscribed rectangle in ellipse $(C)$.

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