According to Wolfram|Alpha, the equation $$a^3+b^3+c^3=d^3$$ has no solutions in the integers. However, this is not the case!

There are, in fact, infinite solutions to this equation, and the minimal solution is quite extraordinary. Where $$a$$, $$b$$, $$c$$ and $$d$$ are positive integers, our minimal solution takes the following form:

$a^2+b^2=c^2$ $a^3+b^3+c^3=d^3$

If $$(a,b,c,d)$$ is our minimal solution, what is $$a+b+c+d$$?

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