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

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