
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
The intersection points of the line $3x-4y+12=0$ with the coordinate axes are the endpoints of a diameter of circle $C$.
(Ⅰ) Find the equation of circle $C$;
(Ⅱ) The length of chord $AB$ of circle $C$ is $\sqrt{21}$, and it passes through the point $(1, \frac{1}{2})$. Find the equation of the line containing chord $AB$.

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