逐步解出上面图中题目---25. (13 分) 已知：抛物线 $C_1$: $y=a(x-h)^2+2h$ ($a \ne 0, h \ge 1$)，其顶点为 $A$，且与 $y$ 轴交于点 $B(0, 1)$，将抛物线 $C_1$ 沿直线 $y=-1$ 翻折，得到抛物线 $C_2$. (1) 当 $h=1$ 时， ①求抛物线 $C_1$ 的解析式，并直接写出顶点 $A$ 的坐标. ②点 $D$ 在抛物线 $C_1$ 上，延长 $AD$ 至 $E$ 使得 $AE=2AD$，若点 $E$ 落在抛物线 $C_2$ 上，求 $D$ 的坐标. (2) 动点 $M$ 在抛物线 $C_1$ 的对称轴 $x=h$ 上 ($M$ 不与 $A$ 重合)，过 $M$ 作直线垂直于 $y$ 轴，交 $C_1$ 于点 $P$ ($P$ 在对称轴左侧)，交 $C_2$ 于点 $Q$ ($Q$ 在对称轴右侧). 当点 $P$ 与点 $B$ 重合时，若 $MQ = \sqrt{3} MP$ 时，求 $h$ 的值. **Chart Description:** The image contains two coordinate plane diagrams side by side. Both are labeled "(备用图)" (For reference). **Left Chart:** * Type: Coordinate plane with a parabola. * Coordinate Axes: X-axis and Y-axis intersecting at the origin O. * Scales: X-axis is marked from -2 to 3 with integer steps. Y-axis is marked from -3 to 3 with integer steps. * Parabola: An upward-opening parabola is shown, passing through approximately (0, 1), (1, 2), (-1, 2). The vertex appears to be at (0, 1). However, this graph does not seem to match the general form $y=a(x-h)^2+2h$ with $h \ge 1$. **Right Chart:** * Type: Coordinate plane with a parabola. * Coordinate Axes: X-axis and Y-axis intersecting at the origin O. * Scales: X-axis is marked from -2 to 3 with integer steps. Y-axis is marked from -3 to 3 with integer steps. * Parabola: A downward-opening parabola is shown, passing through approximately (0, -3), (1, -4), (-1, -4). The vertex appears to be at (0, -3). Note: The charts are likely illustrative examples or reference graphs and may not directly represent the specific parabolas $C_1$ and $C_2$ described in the problem text, especially given the parameters $h \ge 1$.