
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 a circle $M$ that passes through points $A(1,-2)$ and $B(-1,0)$, and the sum of its intercepts on the two coordinate axes is $2$.

$(1)$ Find the equation of circle $M$;

$(2)$ If $P(2, \frac{1}{2})$ is a point inside the circle, find the equation of line $l$ that passes through point $P$ and cuts the shortest chord on circle $M$.

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