\[\Large{\vec{F} = x\hat{\imath} + y\hat{\jmath} + z\hat{k}}\]

If the surface integral of the vector field \(\vec{F}\) over a unit sphere centered on the origin can be expressed as \(\alpha \pi\), determine the value of \(\alpha\).

Note: \(\hat{\imath}, \hat{\jmath},\hat{k}\) are unit vectors associated with the three Cartesian coordinate axes (\(x,y\), and \(z\) respectively).

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