
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 $z=\frac{2+4mi}{1-i}$, where $m\in\mathbb{R}$ and $i$ is the imaginary unit.

$(1)$ If $z$ is a pure imaginary number, find the value of $m$;

$(2)$ Let $\bar{z}$ be the conjugate of $z$. If the point corresponding to the complex number $\bar{z}+2z$ is in the first quadrant of the complex plane, find the range of values for $m$.

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