
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 $xoy$, the parametric equation of curve $C_{1}$ is $\begin{cases} x= \sqrt {3}\cos \alpha \\ y=\sin \alpha \end{cases}$ ($\alpha$ is the parameter), with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. The polar equation of curve $C_{2}$ is $\rho\sin (\theta+ \frac {\pi}{4})=4 \sqrt {2}$.  
$(1)$ Find the ordinary equation of curve $C_{1}$ and the Cartesian coordinate equation of curve $C_{2}$.  
$(2)$ Let $P$ be a moving point on curve $C_{1}$, find the minimum distance from point $P$ to points on curve $C_{2}$, and find the coordinates of point $P$ at this time.

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