
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
Given the curve $C$: $\begin{cases} x=2\cos \alpha \\ y=2+2\sin \alpha\end{cases}$ (where $\alpha$ is the parameter), and the line $l$: $\begin{cases} x=3+ \sqrt {2}t \\ y= \sqrt {2}t\end{cases}$ (where $t$ is the parameter), with the origin as the pole and the non-negative half-axis of $x$ as the polar axis to establish a polar coordinate system.  
$(1)$ Write the polar equation of curve $C$ and the standard equation of line $l$;  
$(2)$ Point $A$ is on curve $C$, and point $B$ is on line $l$. Find the minimum distance $|AB|$ between points $A$ and $B$.

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