
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
Elective 4-4: Coordinate System and Parametric Equations

In the Cartesian coordinate system $xOy$, the curve $C_1$ is given by the parametric equations $\begin{cases}x=\cos \alpha \\ y= \sqrt{3}\sin \alpha\end{cases} (\alpha$ is the parameter). 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 $\rho \sin (\theta +\dfrac{\pi }{4})=2\sqrt{2}$.

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

$(2)$ Let point $P$ be on $C_1$ and point $Q$ be on $C_2$, find the minimum value of $|PQ|$.

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