
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 parametric equation of circle $C$ as $\begin{cases} x=1+3\cos \theta \\ y=3\sin \theta \end{cases}$ (where $\theta$ is the parameter), and establishing a polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of line $l$ is $\theta= \frac {\pi}{4}(\rho\in\mathbb{R})$.  
$(1)$ Write the polar coordinates of point $C$ and the polar equation of circle $C$;  
$(2)$ Points $A$ and $B$ are respectively on circle $C$ and line $l$, and $\angle ACB= \frac {\pi}{3}$. Find the minimum length of segment $AB$.

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