
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 line $L: y = -1$ and the circle $C: (x-0)^2 + (y-2)^2 = 1$, if a moving circle $M$ is tangent to $L$ and externally tangent to the circle $C$, then the equation of the locus of the center $M$ of circle $M$ is __________.

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