
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 in the Cartesian coordinate system $xOy$, the equation of circle $M$ is $(x-4)^2+y^2=1$. With the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, and taking the same unit length as the Cartesian coordinate system, a polar coordinate system is established. The polar equation of line $l$ is $\rho \sin \left( \theta +\frac{\pi }{6} \right)=\frac{1}{2}$.

 (Ⅰ) Find the Cartesian equation of line $l$ and the parametric equation of circle $M$;

 (Ⅱ) Find the minimum distance from a point on circle $M$ to line $l$.

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