
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 that the parametric equation of line $l$ is 
$$
\begin{cases} 
x=m+ \frac{\sqrt{2}}{2}t \\
y= \frac{\sqrt{2}}{2}t
\end{cases}
(t \text{ is the parameter}),
$$
with the origin as the pole and the positive $x$-axis as the initial line of a polar coordinate system, 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 curve $C$ lies on line $l$. 

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

(Ⅱ) Let the perimeter of the inscribed rectangle of curve $C$ be $P$. Find the maximum value of $P$.

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