
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 \(a, b, c\) are distinct positive integers such that
$$
\{a+b, b+c, c+a\}=\{n^{2},(n+1)^{2},(n+2)^{2}\},
$$
where \(n\) is a positive integer. Find the minimum value of \(a^{2} + b^{2} + c^{2}\).

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