
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 a complex number $z=1+mi$ (where $i$ is the imaginary unit, $m\in\mathbb{R}$), and $\overline{z}\cdot(3+i)$ is a pure imaginary number ($\overline{z}$ is the conjugate of $z$).

$(1)$ Let the complex number $z_{1}= \frac{m+2i}{1-i}$, find $|z_{1}|$;

$(2)$ Let the complex number $z_{2}= \frac{a-i^{2017}}{z}$, and the point corresponding to the complex number $z_{2}$ is in the first quadrant, find the range of the real number $a$.

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