这道题怎么做？---**Textual Information:** **Problem Description:** 如图已知三棱柱 $ABC-A_1B_1C_1$ 的底面是正三角形，侧面 $BB_1C_1C$ 是矩形，$M, N$ 分别为 $BC, B_1C_1$ 的中点，$P$ 为 $AM$ 上一点，过 $B_1, C_1$ 和 $P$ 的平面交 $AB$ 于 $E$，交 $AC$ 于 $F$. **Question 1:** (1) 证明：$AA_1 // MN$，且平面 $A_1AMN \perp$ 面 $EB_1C_1F$； **Question 2:** (2) 设 $O$ 为 $\triangle A_1B_1C_1$ 的中心，若 $AO // 面 EB_1C_1F$，且 $AO=AB$，求直线 $B_1E$ 与平面 $A_1AMN$ 所成角的正弦值. **Chart/Diagram Description:** * **Type:** 3D geometric figure, representing a triangular prism. * **Main Elements:** * **Vertices:** Labeled points $A, B, C, A_1, B_1, C_1$. * **Edges:** Lines connecting vertices. Solid lines represent visible edges ($AA_1, A_1B_1, A_1C_1, AB, AC, BC, BB_1, CC_1$). Dashed lines represent hidden edges ($B_1C_1, AB_1, AC_1$). * **Points:** $M$ is the midpoint of $BC$. $N$ is the midpoint of $B_1C_1$. $P$ is a point on the line segment $AM$. $E$ is a point on $AB$. $F$ is a point on $AC$. $O$ is a point in the top face $A_1B_1C_1$, marked as the center. * **Planes:** The base planes are $\triangle ABC$ and $\triangle A_1B_1C_1$. The side faces are rectangles $ABB_1A_1$, $ACC_1A_1$, $BB_1C_1C$. A plane passing through $B_1, C_1$ and $P$ intersects $AB$ at $E$ and $AC$ at $F$. The plane is likely $EB_1C_1F$. Another plane mentioned is $A_1AMN$. * **Lines:** Line segments connecting the points defined, including $AM$, $MN$, $A_1N$, $A_1M$, $B_1E$, $C_1F$, $EF$. There is a line $AO$ shown, connecting $A$ to $O$. * **Relative Position and Direction:** The figure shows a triangular prism $ABC-A_1B_1C_1$. $A_1$ is vertically above $A$, $B_1$ above $B$, and $C_1$ above $C$. The base $\triangle ABC$ is in the bottom plane. The base $\triangle A_1B_1C_1$ is in the top plane. $M$ is on $BC$, $N$ is on $B_1C_1$, $P$ is on $AM$. $E$ is on $AB$, $F$ is on $AC$. $O$ is in the plane $A_1B_1C_1$. The plane $EB_1C_1F$ is shown cutting through the prism. The plane $A_1AMN$ is also implicitly defined by the points. * **Annotations:** Points are labeled with letters ($A, B, C, A_1, B_1, C_1, M, N, P, E, F, O$).