
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
In the Cartesian coordinate system, the center of circle $C$ lies on the line $x-y=0$, and circle $C$ passes through point $P(2,0)$ and point $Q(-1, \sqrt{3})$.  
$(1)$ Find the standard equation of circle $C$.  
$(2)$ Find the equation of the line passing through point $M(2,1)$ and intersecting circle $C$ at exactly one point.

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