
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 plane rectangular coordinate system $xOy$, the circle $C$ passes through the points $A(1,0)$ and $B(-1,2)$, and the center of the circle lies on the line $2x-y+2=0$.  
$(1)$ Find the standard equation of the circle $C$.  
$(2)$ If the line $x=ay+3$ intersects the circle $C$ and forms a chord of length $2\sqrt{3}$, find the value of the real number $a$.

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