
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 center of circle $C$ lies on the $x$-axis and passes through points $A(-1,0)$ and $B(1,2)$.
$(1)$ Find the standard equation of circle $C$.
$(2)$ A line $l$ passing through point $P(0,2)$ intersects circle $C$ at points $M$ and $N$, where $|MN|=2\sqrt{3}$. Find the equation of line $l$.

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