
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 $z=(m^{2}-8m+15)+(m^{2}-4m+3)i$, where $m\in R$. If $z$ is a purely imaginary number, find the value of the real number $m$;<br/>$(2)$ Given a complex number $z=\frac{{2+4mi}}{{1+i}}$ ($m\in R$, $i$ is the imaginary unit). Let $\overline{z}$ be the conjugate of $z$, and the point corresponding to the complex number $\overline{z}$ on the complex plane is in the first quadrant. Find the range of values for $m$.

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