
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 $\frac{a}{b}=\frac{c}{d}=\frac{e}{f}=3\left(b+d+f\neq 0\right)$, and $k=\frac{{a+c+e}}{{b+d+f}}$.  
$(1)$ Find the value of $k$;  
$(2)$ If $x_{1}$ and $x_{2}$ are the two roots of the equation $x^{2}-3x+k-2=0$, find the value of $x_{1}^{2}+x_{2}^{2}$.

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