
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 a circle $C$ that passes through points $A(2,0)$ and $B(1,- \sqrt {3})$, and the center of the circle $C$ is on the line $y=x$.
$(1)$ Find the equation of circle $C$;
$(2)$ A line $l$ passing through the point $(1, \frac { \sqrt {3}}{3})$ intercepts the circle, and the length of the chord is $2 \sqrt {3}$. Find the equation of line $l$.

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