
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 $xOy$, with the origin $O$ as the pole and the non-negative half-axis of $x$ as the polar axis, a polar coordinate system is established. The parametric equation of curve $C_{1}$ is $\begin{cases} x=2 \sqrt {2}\cos \alpha \\ y=2\sin \alpha \end{cases} (\alpha \in R, \alpha$ is the parameter), and the polar equation of curve $C_{2}$ is $\rho\cos \theta- \sqrt {2}\rho\sin \theta-5=0$.
$(1)$ Find the general equation of curve $C_{1}$ and the Cartesian coordinate equation of curve $C_{2}$;
$(2)$ Let $P$ be a point on curve $C_{1}$, and $Q$ a point on curve $C_{2}$, find the minimum value of $|PQ|$.

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