
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
Assume a complex number $z=a+bi$ (where $a, b \in \mathbb{R}, a>0$, and $i$ is the imaginary unit) such that $|z| = \sqrt{10}$. The complex number $(1+2i)z$ corresponds to a point on the complex plane that lies on the angle bisector of the first and third quadrants.
(1) Find the complex number $z$;
(2) If $\overline{z} + \frac{m-i}{1+i}$ is a pure imaginary number (where $m \in \mathbb{R}$ and $\overline{z} = a-bi$), find the value of the real number $m$.

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