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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