
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
Let $\overset{ .}{z}$ be the conjugate of the complex number $z$, satisfying $|z- \overset{ .}{z}|=2 \sqrt {3}$.  
(1) If $z$ is a pure imaginary number, find $z$;  
(2) If $z- \overset{ .}{z}^2$ is a real number, find $|z|$.

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