
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 on a plane, establish a polar coordinate system with $O$ as the pole and the positive semi-axis of $x$ as the polar axis. The parametric equations of the curve $C$ are $\begin{cases} x=1+\cos \alpha \\ y=\sin \alpha \end{cases} (\alpha \text{ is the parameter, } \alpha \in \left[ 0,\pi \right])$, and the polar equation of the line $l$ is $\rho = \frac{4}{\sqrt{2}\sin \left( \theta - \frac{\pi }{4} \right)}$.
(I) Write the Cartesian equation of curve $C$ and the polar equation of line $l$.
(II) Let $P$ be any point on curve $C$ and $Q$ be any point on line $l$. Find the minimum value of $|PQ|$.

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