
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 triples  $(a,b,c)$  of positive integers such that
(i)  $a \leq b \leq c$ ;
(ii)  $\text{gcd}(a,b,c)=1$ ; and
(iii)  $a^3+b^3+c^3$  is divisible by each of the numbers  $a^2b, b^2c, c^2a$ .

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