
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 plane rectangular coordinate system $xOy$, the parameter equation of the line $l$ is $\left\{\begin{array}{l}x=-t\\ y=\sqrt{3}-t\end{array}\right.$ ($t$ is the parameter). Taking $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis, the polar coordinate equation of curve $C_{1}$ is $\rho =4\sin \theta$.  
$(1)$ Find the general equation of line $l$ and the rectangular coordinate equation of curve $C_{1}$.  
$(2)$ In the rectangular coordinate system, if the graph of curve $C_{1}$ is shifted down by $2$ units, then the abscissa remains unchanged, and the ordinate is compressed to $\frac{1}{2}$ of the original, resulting in curve $C_{2}$. Line $l$ intersects curve $C_{2}$ at points $M$ and $N$, and intersects the $x$-axis at point $P$. Find the value of $\frac{1}{|PM|}+\frac{1}{|PN|}$.

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