
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
On a board, the numbers \(a\), \(b\), and \(c\) were written. They were erased and replaced by the numbers \(a^{2} + 2bc\), \(b^{2} + 2ca\), and \(c^{2} + 2ab\). It turned out that the numbers written on the board were the same as initially, possibly in a different order. Find all possible values of the sum \(a + b + c\).

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