
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 center of circle $C$ lies on the $x$-axis and passes through points $A(-1, 1)$ and $B(1, 3)$, find the equation of circle $C$. Additionally, let line $y = -x + m$ intersect circle $C$ at points $M$ and $N$. Determine whether a circle with diameter $MN$ can pass through the origin. If it can, find the equation of line $MN$; otherwise, explain the reason.

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