
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
Real numbers \( a, b, c \) satisfy \( a^{2} + b^{2} + c^{2} = \lambda (\lambda > 0) \). Find
\[ 
f = \min \left\{ (a - b)^{2}, (b - c)^{2}, (c - a)^{2} \right\} 
\]
and its maximum value.

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