Due to an error, everyone will receive free points for this problem.
An octahedron has vertices at \((\pm 1, 0,0)\), \((0, \pm 1, 0)\), and \((0,0 \pm 1)\). The notation \((\pm 1, 0, 0)\) represents both points \((1,0,0)\) and \((-1,0,0)\). It is rotated about the line \(x = y = z\) by \(120\) degrees. Let \(V\) be the volume of space that the octahedron, including its interior, traces out in this process. Then \(V\) can be written in the form \(\left(\frac{a\sqrt{b} + c}{d} \right)\pi\) where \(b\) is not divisible by the square of any prime, and \(\gcd(a,c,d) = 1\). Find \(a + b + c + d\).