
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 parametric equations of curve $C_{1}$ are $\left\{\begin{array}{l}x=3+2\sqrt{2}\cos\alpha,\\ y=2\sqrt{2}\sin\alpha\end{array}\right.$ ($\alpha$ is the parameter). Establish a polar coordinate system with the origin $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis. The polar coordinate equation of curve $C_{2}$ is $\rho \sin^{2}\theta -6\cos\theta =0$.  
$(1)$ Find the Cartesian equation of curve $C_{1}$ and the rectangular coordinate equation of curve $C_{2}$;  
$(2)$ Let the line $l$ be $\left\{\begin{array}{l}x=3+\frac{1}{2}t\\ y=\frac{{\sqrt{3}}}{2}t\end{array}\right.$ ($t$ is the parameter), and the intersection points of curves $C_{2}$ and $C_{1}$ from top to bottom are $P$, $M$, $N$, $Q$. Find the value of $|PM|+|NQ|$.

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