
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 parametric equation of curve $C_{1}$ is: 
$$
\begin{cases} 
x=2\cos \alpha \\ 
y=\sqrt{2}\sin \alpha 
\end{cases}
(\alpha \text{ is the parameter})
$$
Taking the origin as the pole and the positive half-axis of $x$ as the polar axis, and using the same length unit as the Cartesian coordinate system, a polar coordinate system is established. The polar equation of curve $C_{2}$ is: 
$$
\rho =\cos \theta
$$

$(1)$ Find the Cartesian coordinate equation of curve $C_{2}$;

$(2)$ If $P$ and $Q$ are any points on curves $C_{1}$ and $C_{2}$ respectively, find the minimum value of $|PQ|$.

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