这个题给我证明一下吧，用中文，并生成尽量详细的视频---**Question 1** **Textual Information:** * **Question Stem:** 如图, 正方体 $ABCD-A_1B_1C_1D_1$ 的棱长为 $a$, 点 $P$ 是棱 $AD$ 上一点, 且 $AP=\frac{a}{3}$, 过 $B_1$, $D_1$, $P$ 的平面交底面 $ABCD$ 于 $PQ$, $Q$ 在直线 $CD$ 上, 则 $PQ=$__________. * **Mathematical Formulas:** $AP = \frac{a}{3}$ * **Units/Labels:** 棱长 (edge length) is $a$. **Chart/Diagram Description:** * **Type:** 3D geometric figure (Cube). * **Main Elements:** * **Shape:** A cube with vertices labeled A, B, C, D on the top face and A1, B1, C1, D1 on the bottom face. The figure is oriented such that A is in the front-left, B front-right, C back-right, D back-left on the top. A1 is directly below A, B1 below B, C1 below C, and D1 below D on the bottom face. * **Points:** Vertices are labeled A, B, C, D, A1, B1, C1, D1. Point P is marked on edge AD. * **Lines:** Edges of the cube are shown. Edges AB, BC, CD, DA, AA1, BB1, CC1, DD1, A1B1, B1C1, C1D1, D1A1. Solid lines represent visible edges (AB, BC, CD, DA, AA1, BB1, CC1). Dashed lines represent hidden edges (A1B1, B1C1, C1D1, D1A1, DD1). A solid line segment from P to Q (not explicitly drawn but implied by the problem statement "交底面 ABCD 于 PQ"). * **Labels and Annotations:** Vertices are labeled A, B, C, D, A1, B1, C1, D1. Point P is labeled on edge AD. * **Relative Position and Direction:** The base of the cube is A1B1C1D1 and the top is ABCD. The edges AA1, BB1, CC1, DD1 are vertical. P is a point on the edge AD. The plane passing through B1, D1, and P intersects the base plane ABCD along the line PQ. Q is on the line CD.