
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 equations of curve $C_{1}$ are $\left\{\begin{array}{l}x=\sin\alpha\\ y=\sqrt{3}\cos\alpha\end{array}\right.$ ($\alpha$ is the parameter). Establish a polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis. The polar coordinate equation of curve $C_{2}$ is $ρ\cos(θ+\frac{π}{4})=2\sqrt{2}$.  
$(1)$ Write down the Cartesian equation of $C_{1}$ and $C_{2}$.  
$(2)$ Let point $P$ be on $C_{1}$ and point $Q$ be on $C_{2}$. Find the minimum value of $|PQ|$ and the Cartesian coordinates of $P$ at that point.

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