
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 the real roots of the equation \( x^{2}-(k-2)x+(k^{2}+3k+5)=0 \) (where \( k \) is a real number), find the maximum value of \( x_{1}^{2} + x_{2}^{2} \).

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