给出这道题的解答视频，用中学生对应年级的知识---**Textual Information:**
如图, 中心在原点O的椭圆$\Gamma$的右焦点为$F(2\sqrt{3}, 0)$, 长轴长为8. 椭圆$\Gamma$上有两点$P$、$Q$, 连接$OP$、$OQ$, 记它们的斜率分别为$k_{OP}$、$k_{OQ}$, 且满足$k_{OP} \cdot k_{OQ} = -\frac{1}{4}$.
**(1) 简答题**
求椭圆$\Gamma$的标准方程;
**(2) 简答题**
求证: $|OP|^2 + |OQ|^2$为一定值, 并求出这个定值;
**(3) 简答题**
设直线$OQ$与椭圆$\Gamma$的另一个交点为$R$, 直线$RP$和$PQ$分别与直线$x = 4\sqrt{3}$交于点$M$、$N$, 若$\triangle PQR$和$\triangle PMN$的面积相等, 求点$P$的横坐标.
**Chart/Diagram Description:**
* **Type:** Geometric figure illustrating an ellipse, points, lines, and coordinate axes.
* **Main Elements:**
* **Coordinate Axes:** Horizontal X-axis and vertical Y-axis intersecting at the origin O. Arrows indicate the positive direction of both axes. Labels 'x' and 'y' are present near the ends of the respective axes.
* **Origin:** Point O at the intersection of the X and Y axes.
* **Ellipse:** An ellipse $\Gamma$ is centered at the origin O. Its major axis lies along the X-axis.
* **Points:**
* P: A point on the ellipse in the first quadrant.
* Q: A point on the ellipse in the fourth quadrant.
* R: A point on the ellipse in the second quadrant, collinear with O and Q. It appears to be the point diametrically opposite to Q.
* M: A point on a vertical line to the right of the ellipse, intersection of line RP and the vertical line.
* N: A point on the same vertical line to the right of the ellipse, intersection of line PQ and the vertical line.
* **Lines:**
* Line segment OP: Connects the origin O to point P.
* Line segment OQ: Connects the origin O to point Q.
* Line segment OR: Connects the origin O to point R. Points R, O, Q appear collinear.
* Line segment RP: Connects point R to point P.
* Line segment PQ: Connects point P to point Q.
* A vertical line labeled $x=4\sqrt{3}$ (implied by the problem description and its position relative to the ellipse) is shown intersecting the X-axis at some point to the right of the ellipse's right vertex. Points M and N lie on this line.
* **Relative Position and Direction:** The ellipse is centered at O. P is in the first quadrant, Q in the fourth, and R in the second. The vertical line $x=4\sqrt{3}$ is to the right of the ellipse. Lines RP and PQ intersect this vertical line at M and N respectively. Triangles PQR and PMN are formed.