A line \(L_1\) passing through the point \(A\) with position vector \( \vec{a} = 4\hat{i} + 2\hat{j} + 2\hat{k} \) is parallel to the vector \( \vec{b} = 2\hat{i} + 3\hat{j} + 6\hat{k} \). The length of the perpendicular drawn from a point \(P\) with position vector \( \vec{p} = \hat{i} + 2\hat{j} + 3\hat{k} \) to \(L_1\) is