
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 $x$, $x^4 + 2x^3 + (3+k)x^2 + (2+k)x + 2k = 0$ has real roots, and the product of all real roots is $-2$, find the sum of the squares of all real roots.

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