
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
Find all  $a,b,c \in \mathbb{N}$  such that 
\[a^2b|a^3+b^3+c^3,\qquad b^2c|a^3+b^3+c^3, \qquad c^2a|a^3+b^3+c^3.\]

[PS: The original problem was this:
Find all  $a,b,c \in \mathbb{N}$  such that 
\[a^2b|a^3+b^3+c^3,\qquad b^2c|a^3+b^3+c^3, \qquad \color{red}{c^2b}|a^3+b^3+c^3.\]
But I think the author meant  $c^2a|a^3+b^3+c^3$ , just because of symmetry]

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