
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 that the slope of line $l$ is $\alpha$, $\cos\alpha=-\frac{1}{2}$, and this line passes through point $A\left(\frac{\sqrt{3}}{3},2\right)$.  
$(1)$ Find the equation of line $l$.  
$(2)$ Line $kx-y+1-\sqrt{3}k=0$ always passes through a fixed point $B$. Find the distance from point $B$ to line $l$.

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