
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]


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In the rectangular coordinate system $xoy$, the parametric equations of curve $C_{1}$ are given by $x = \sqrt{3} \cos \alpha$, $y = \sin \alpha$ (where $\alpha$ is the parameter), and curve $C_{2}$ has the polar coordinate equation $\rho \sin(\theta + \frac{\pi}{4}) = 2\sqrt{2}$ with the origin as the pole and the positive half of the $x$-axis as the polar axis.

(I) Write the Cartesian equation for $C_{1}$ and the rectangular coordinate equation for $C_{2}$.
(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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