Please reason step by step, and put your final answer within \boxed{}.
Given a cube \(ABCD A_{1} B_{1} C_{1} D_{1}\), a plane passing through \(A\) and tangent to the sphere inscribed in the cube intersects the edges \( A_{1} B_{1} \) and \( A_{1} D_{1} \) at points \( K \) and \( N \) respectively. Determine the measure of the dihedral angle between the planes \( AC_{1}K \) and \( AC_{1}N \).
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