
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
Let $z$ be a complex number.  
$(1)$ If $\frac{25}{z}=\frac{-4-3i}{i}$, find the value of $|z|$.  
$(2)$ If the real-coefficient quadratic equation $x^{2}+kx+k=0$ has two complex roots $x_{1}$ and $x_{2}$, and ${x}_{1}^{2}+{x}_{2}^{2}=3$, find the value of the real number $k$.

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