
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 curve \(C\) is given by the parametric equations \( \begin{cases} x=1+4\cos \theta \\ y=2+4\sin \theta \end{cases} \) (where \(\theta\) is the parameter). A line \(l\) passes through a fixed point \(P(3,5)\) and has an inclination angle of \( \frac{\pi}{3} \).
(1) Provide the parametric equations for line \(l\) and the standard (Cartesian) equation for the curve \(C\);
(2) Suppose line \(l\) intersects curve \(C\) at two points \(A\) and \(B\). Calculate the value of \(|PA|\cdot|PB|\).

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