A Chord of Length Two

The ellipse $\Gamma:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) passes through the point $P(0,\sqrt 3)$; its left and right vertices are $A_1$, $A_2$, and the product of the slopes of $PA_1$ and $PA_2$ is $-\dfrac 34$. A moving point $Q(m,n)$ lies on $\Gamma$ with $m<0$ and $n>0$. The line $l:3mx+4ny=0$ meets $\Gamma$ in the first quadrant at a point $R$. A point $T$ lies on the segment $OR$ ($O$ the origin) with $|QT|=2$.

1. Find the equation of the ellipse $\Gamma$. 2. Prove that the line $QT$ passes through a fixed point $S$, and find the coordinates of $S$.