
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 rectangular coordinate system $(xOy)$, the parametric equations of curve $(C_1)$ are given by $\begin{cases} x = \sqrt{3}\cos\alpha \\ y = \sin\alpha \end{cases}$ ($\alpha$ is a parameter), and the polar coordinate equation of curve $(C_2)$ is $\rho\sin\left(\theta + \frac{\pi}{4}\right) = 2\sqrt{2}$ with the coordinate origin as the pole and the positive semi-axis of $x$ as the polar axis.
(I) Write the Cartesian 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 $P$ in this case.

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