Number theory at its best

Assume \(a,b\) and \(c\) are three positive integers that satisfy the conditions:

  • \(a+b=c+12\)
  • \((a^2+b^2)^{1/3} =c+2\)
  • \(c,b+4,2a+7\) are in a geometric progression.

Find the value of \(a+b+c\).

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