In a cuboid made of glass there are 12 spheres which perfectly fit in the cuboid in \(3 \times 2 \times 2 \) orientation. All the spheres are identical. The dimensions of the cuboid \( (l:b:h) \) are in \(3:2:2\) ratio. The cuboid has an inner volume \(V_c\), the sphere has an outer volume \(V_s\), inner volume \(v_s\), outer radius \(R\), inner radius \(r\) and width \(w\). The volume of the empty spaces in the cuboid *(i.e. difference between the volume of the cuboid and the volumes of all the spheres)* is equal to the inner volume of 30 such spheres. If the outer radius of a sphere is \(x\), find \(r\) in terms of \(x\).

If for positive integers \(a,b,c,d,e\), we have \( r = x \left (\sqrt[e]{ \frac{a}{b \pi} - \frac{c}{d} } \right) \), where \( \text{gcd}(a, b)=\text{gcd}(c,d)=1\), with \(e\) minimized. Find the value of \(a+b+c+d+e\)

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