
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 the general equation of a conic section and two points: \( M(x_{0}, y_{0}) \) and \( N(x', y') \). Assuming that point \( M \) is fixed, what geometric locus does \( N \) describe if, for any position of \( M N \), it intersects the conic section at two coincident points?

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