
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 right triangle $ABC$ with vertices $A(1,2)$, $B(-1,-2)$, and $C(1,-2)$, circle $E$ is the circumcircle of $\triangle ABC$.
- $(I)$ Find the equation of circle $E$;
- $(II)$ Find the shortest chord length intercepted by the line $lmx-y-m+1=0$ on circle $E$ and the equation of the corresponding line $l$.

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