
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 the complex number $z= \frac {(1-i)^{2}+3(1+i)}{2-i}$  
(1) If $z\cdot(m+2i)$ is a pure imaginary number, find the value of the real number $m$;  
(2) If the complex number $z_1$ and $z$ correspond to points in the complex plane that are symmetric about the imaginary axis, find the real part of $z_1$;  
(3) If the complex number $z_2=a+bi$ ($a,b\in\mathbb{R}$), and $z^2+az+b=1-i$, find $|z_2|$

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