
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 line $l$ passing through point $(0,1)$ that intersects the curve $C: y = x + \frac{1}{x} \, (x > 0)$ at two distinct points $M$ and $N$, find the locus of the intersection points of the tangents to the curve $C$ at points $M$ and $N$.

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