
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 line $l$ is $\begin{cases}x= \frac{ \sqrt{2}}{2}t \\ y=3+ \frac{ \sqrt{2}}{2}t\end{cases} (t$ is the parameter$)$, in the polar coordinate system with $O$ as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of curve $C$ is $\rho =4\sin \theta -2\cos \theta $

$(1)$ Find the general equation of line $l$ and the Cartesian coordinate equation of curve $C$;

$(2)$ If the intersection of line $l$ and the $y$-axis is point $P$, and the intersections of line $l$ and curve $C$ are points $A,B$, find the value of $\frac{1}{|PA|}+\frac{1}{|PB|}$.

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