
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
(1) Given a complex number $z$ whose corresponding point in the complex plane is in the fourth quadrant, and $|z|=1$, also $z + \overline{z} = 1$, find $z$.

(2) Given a complex number $z = \frac{5m^{2}}{1-2i} - (1+5i) - 3(2+i)$ is a pure imaginary number, find the value of the real number $m$.

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