
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
The center of circle $C$ is $C(2,0)$, and it passes through point $A(\frac{3}{2},\frac{\sqrt{3}}{2})$.  
$(1)$ Find the standard equation of circle $C$.  
$(2)$ Line $l:kx+y+1=0$ intersects circle $C$ at points $M$ and $N$, and $|MN|=\sqrt{2}$. Find the value of $k$.

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