
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 $l$ passes through the point $(3,2)$ and is perpendicular to the line $y=-\frac{7}{2}x+1$, the center of the circle $C$ lies on the line $l$ and passes through the points $A(6,0)$ and $B(1,5)$.  
$(1)$ Find the equation of the line $l$.  
$(2)$ Find the standard equation of the circle $C$.

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