
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 line $l_{1}$ passes through the origin and is parallel to $l_{2}: 3x-2y-1=0$.
$(1)$ Find the equation of the line $l_{1}$;
$(2)$ Find the distance between $l_{1}$ and $l_{2}$;
$(3)$ If a circle $C$ passes through the points $A(1,3)$ and $B(2,2)$, and is bisected by the line $l_{1}$, find the equation of the circle $C$.

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