The triplets of consecutive integers $(0,1,2)$ and $(1,2,3)$ have the property that the sum of their cubes is a perfect square:

$0^3 + 1^3 + 2^3 = 3^2,\quad 1^3 + 2^3 + 3^3 = 6^2 .$

Find the next smallest triplet of consecutive integers $(a, b, c)$ that has this property.

What is $a+b+c?$

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