
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 $i$ is the imaginary unit, the conjugate of a complex number $z$ is $\overline{z}$, and it satisfies $z+2\overline{z}=3-2i$.  
$(1)$ Find the modulus $|z|$ of the complex number $z$;  
$(2)$ If the point corresponding to the complex number $z(2-mi)$ in the complex plane is in the second quadrant, find the range of real number $m$.

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