
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 $\begin{cases}x= \sqrt{3}\cos \theta \\ y=\sin \theta\end{cases}$ ($\theta$ is the parameter), and in the polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of curve $C_2$ is $\rho\sin \left(\theta+ \frac{\pi}{4}\right)=2 \sqrt{2}$.

$(1)$ Write the general equation of $C_1$ and the Cartesian coordinate equation of $C_2$;

$(2)$ Suppose point $P$ is on $C_1$, and point $Q$ is on $C_2$, find the minimum value of $|PQ|$ and the Cartesian coordinates of $P$ at this time.

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