
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 real numbers \( x, y, z \) that satisfy \( x + y = z - 1 \) and \( xy = z^2 - 7z + 14 \), what is the maximum value of \( x^2 + y^2 \)? For which value of \( z \) does \( x^2 + y^2 \) attain its maximum value?

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