
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 $\begin{cases} x=\cos\alpha \\ y=\sqrt{3}\sin\alpha \end{cases}$ (where $\alpha$ is a parameter). In the polar coordinate system established with the coordinate origin as the pole and the positive semi-axis of $x$ as the polar axis, the polar equation of curve $C_2$ is given by $\rho\cos(\theta+\frac{\pi}{4})=2\sqrt{2}$.
(I) Write the Cartesian equation of $C_1$ and the rectangular coordinate equation of $C_2$.
(II) Suppose point $P$ lies on $C_1$ and point $Q$ lies 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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