
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 
$$
\begin{cases}
x = \sqrt{3} \cos θ \\
y = \sin θ
\end{cases} 
(θ \text{ is the parameter}),
$$
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 $\rho \sin \left(\theta + \frac{π}{4}\right) = 2\sqrt{2}$.

$(I)$ Write down the general equation of $C_{1}$ and the rectangular coordinate equation of $C_{2}$;

$(II)$ Let point $P$ be on $C_1$ and point $Q$ be on $C_2$, find the minimum value of $|PQ|$ and the rectangular coordinates of $P$ at this time.

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