
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 equation in terms of $x$: $x^{2}-2kx+k^{2}-k-1=0$ has two distinct real roots $x_{1}$ and $x_{2}$.  
$(1)$ If $k=5$, find the value of ${x_1}x_2^2+x_1^2{x_2}$;  
$(2)$ If $x_{1}-3x_{2}=2$, find the value of the real number $k$.

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