
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, the curve $C_{1}: x^{2}+y^{2}=1$ undergoes a scaling transformation $\begin{cases} x'=2x \\ y'=y \end{cases}$ to obtain the curve $C_{2}$. With the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of the curve $C_{3}$ is $\rho=-2\sin \theta$.  
$(1)$ Find the parametric equations of curves $C_{2}$ and $C_{3}$.  
$(2)$ If $P$ and $Q$ are moving points on curves $C_{2}$ and $C_{3}$ respectively, find the maximum value of $|PQ|$.

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