
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
In a 3D Cartesian coordinate system, given points A(3, 0, 1) and B(1, 0, -3), determine:  
(1) Whether there exists a point M on the y-axis such that $|MA| = |MB|$?  
(2) Whether there exists a point M on the y-axis such that $\triangle MAB$ is an equilateral triangle? If there is such a point M, find its coordinates.

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