
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
A circle C, whose center is on the line $2x-y-7=0$, intersects the y-axis at two points A(0, -4) and B(0, -2).  
(1) Find the equation of circle C;  
(2) If the line $l: kx-y+k=0$ is tangent to circle C, find the value of the real number $k$;  
(3) Find the equation of the circle that is symmetric to circle C with respect to the line $l_1: y=2x+1$.

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