
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
Determine the maximum value of \( m^2 + n^2 \) where \( m \) and \( n \) are integers satisfying
\[ m, n \in \{1, 2, \ldots, 100\} \quad \text{and} \quad \left(n^2 - mn - m^2\right)^2 = 1 \]

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