
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 $z$ is a complex number, and $i$ is the imaginary unit, both $z+3+4i$ and $\frac{z}{1-2i}$ are real numbers.  
(1) Find the complex number $z$;  
(2) If the complex number $(z-mi)^2$ corresponds to a point in the second quadrant on the complex plane, find the range of the real number $m$.

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