
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 $x$, $x^2 + 2kx + k^2 + k + 3 = 0$, with roots $x_1$ and $x_2$, find the minimum value of $(x_1 - 1)^2 + (x_2 - 1)^2$.

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