
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 that $z$ is a complex number, both $z+2i$ and $\frac{z}{2-i}$ are real numbers, where $i$ is the imaginary unit.  
(1) Find the complex number $z$ and its modulus $|z|$;  
(2) If $z_1 = \overline{z}+3m+(m^2−6)i$ lies in the fourth quadrant, determine the range of values for the real number $m$.

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