
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 $l$ is $\begin{cases} x=m+ \dfrac {1}{2}t \\ y= \dfrac { \sqrt {3}}{2}t\end{cases}$ (where $t$ is a parameter), and establishing a polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of ellipse $C$ is $5\cos ^{2}\theta+9\sin ^{2}\theta= \dfrac {45}{\rho ^{2}}$. Also, line $l$ passes through the right focus $F$ of ellipse $C$.
$(1)$ Find the maximum value of the area of rectangle $PMNQ$ inscribed in ellipse $C$;
$(2)$ If line $l$ intersects ellipse $C$ at points $A$ and $B$, find the value of $|FA|\cdot|FB|$.

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