
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
Let \( x_{1}, x_{2} \) be the real roots of the equation \( x^{2}-(k-2) x+\left(k^{2}+3 k+5\right)=0 \), 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}
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