
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 that $x_{1}$ and $x_{2}$ are two real roots of the one-variable quadratic equation $x^{2}-6x+k=0$, and (choose one of the conditions $A$ or $B$ to answer the following questions).<br/>$A$: $x_1^2x_2^2-x_1-x_2=115$;<br/>$B$: $x_1^2+x_2^2-6x_1-6x_2+k^2+2k-121=0$.<br/>$(1)$ Find the value of $k$;<br/>$(2)$ Solve this equation.

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