
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
Elective 4-4: Parametric Equations and Coordinate Systems

With the origin $O$ of the Cartesian coordinate system as the pole and the positive half-axis of the $x$-axis as the polar axis, a polar coordinate system is established. It is known that the Cartesian coordinates of point $P$ are $(1,2)$, and the polar coordinates of point $M$ are $(3, \frac{\pi}{2})$. If line $l$ passes through point $P$ and has an inclination angle of $\frac{\pi}{6}$, and circle $C$ has $M$ as its center and a radius of $3$.

$(1)$ Find the parametric equation of line $l$ and the polar equation of circle $C$;

$(2)$ Suppose line $l$ intersects circle $C$ at points $A$ and $B$, calculate $|PA|\cdot|PB|$.

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