
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
In the Cartesian coordinate system $xOy$, the parametric equation of circle $C$ is 
$$
\begin{cases}
x=4\cos \theta, \\
y=4\sin \theta
\end{cases}
$$
(where $\theta$ is the parameter). The line $l$ passes through the point $P(1,2)$, and the angle of inclination $\alpha$ is $\frac{\pi}{6}$.

(1) Write the standard equation of circle $C$ and the parametric equation of line $l$;

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

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