
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 equation $x^2 + y^2 - 2(m + 3)x + 2(1 - 4m^2)y + 16m^4 + 9 = 0$ represents a circle.
(1) Find the range of the real number $m$; (2) Find the range of the radius $r$ of the circle; (3) Find the equation of the trajectory of the circle center.

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