
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 following conditions:①${z_2}\overline{{z_2}}=10(a＞0)$; ②The point representing $\frac{{{z_1}}}{{{z_2}}}$ on the complex plane lies on the line $x+2y=0$; ③$z_{1}(a-i) \gt 0$. Choose one of the three conditions to fill in the blank below and answer the following questions:<br/>Given complex numbers $z_{1}=1+i$ and $z_{2}=a+3i\left(a\in R\right)(i$ is the imaginary unit), satisfying ____.<br/>$(1)$ If $z=\frac{1}{{{z_1}}}+\frac{1}{{{z_2}}}$, find the complex number $z$ and $|z|$;<br/>$(2)$ If $z_{2}$ is a root of the real coefficient quadratic equation $x^{2}+mx+4-3m=0$, find the value of the real number $m$.

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