
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 $x$: $x^{2}+\left(2k-1\right)x+k^{2}-1=0$ has two real roots $x_{1}$ and $x_{2}$, determine the possible values of the real number $k$ such that ${x}_{1}^{2}+{x}_{2}^{2}=16+{x}_{1}{x}_{2}$.

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