
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 polar coordinate system, the polar coordinates of point $P$ are $(\sqrt{3}, \frac{\pi}{2})$, and the polar equation of curve $C$ is $\rho=4\cos(\theta- \frac{\pi}{3})$. Establish a Cartesian coordinate system with the pole as the origin and the positive half-axis of the $x$-axis as the polar axis. A line $l$ with a slope of $-1$ passes through point $P$.
$(1)$ Write the parametric equation of line $l$ and the Cartesian equation of curve $C$;
$(2)$ If line $l$ intersects curve $C$ at two points $A$ and $B$, find the value of $\frac{|PA|}{|PB|}+ \frac{|PB|}{|PA|}$.

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