如图，直四棱柱 \(ABCD - A_1B_1C_1D_1\) 的底面是菱形，\(AA_1 = 4\)，\(AB = 2\)，\(\angle BAD = 60^\circ\)，\(E\)，\(M\)，\(N\) 分别是 \(BC\)，\(BB_1\)，\(A_1D\) 的中点. （1）证明：\(MN\parallel\) 平面 \(C_1DE\)； （2）求点 \(C\) 到平面 \(C_1DE\) 的距离.---19. (12 分) **Question Stem:** 如图，直四棱柱 $ABCD-A_1B_1C_1D_1$ 的底面是菱形， $AA_1=4, AB=2, \angle BAD=60^\circ$, $E, M, N$ 分别是 $BC, BB_1, A_1D$ 的中点。 **Geometric Figure Description:** Type: 3D geometric figure, a right quadrangular prism with a rhombus base. Main Elements: * Vertices: Labeled $A, B, C, D$ on the bottom base, and $A_1, B_1, C_1, D_1$ on the top base. * Edges: Represents the framework of the prism. Visible edges are shown as solid lines ($A_1A, A_1B_1, A_1D_1$ - top face edges may be partially visible, $AB, BC, CD, DA$ - bottom face edges, $AA_1, BB_1, CC_1, DD_1$ - vertical edges). Hidden edges and internal segments are shown as dashed lines ($A_1B, A_1C, A_1D, B_1C_1, B_1D_1, C_1D_1, A_1B_1, MN, MD, ND, ME, NE, DE$). * Base: The base $ABCD$ is a rhombus. Sides of the rhombus are $AB, BC, CD, DA$. * Height: The height of the prism is represented by the length of the vertical edges, such as $AA_1$. * Points: $E$ is a point on edge $BC$. $M$ is a point on edge $BB_1$. $N$ is a point on the diagonal $A_1D$ of the side face $A_1ADD_1$. The text states $E, M, N$ are midpoints of $BC, BB_1, A_1D$ respectively. * Lines/Segments: Several segments are drawn connecting the vertices and points, including $MN$, $DE$, $C_1D$, $C_1E$, forming the plane $C_1DE$. **Given Information:** * Prism $ABCD-A_1B_1C_1D_1$ is a right quadrangular prism. * Base $ABCD$ is a rhombus. * $AA_1=4$ (height). * $AB=2$ (side length of the rhombus). * $\angle BAD=60^\circ$. * $E$ is the midpoint of $BC$. * $M$ is the midpoint of $BB_1$. * $N$ is the midpoint of $A_1D$. **Sub-questions:** (1) 证明：$MN//$ 平面 $C_1DE$; (Prove: $MN//$ plane $C_1DE$;) (2) 求点 $C$ 到平面 $C_1DE$ 的距离. (Find the distance from point $C$ to plane $C_1DE$.)