
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 $x^2 - 2(k-1)x + 2k^2 - 12k + 17 = 0$, with roots $x_1$ and $x_2$, find the maximum and minimum values of $x_1^2 + x_2^2$ and the roots of the equation at these values.

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