Perpendicular Tangents From a Circle

The ellipse $\Gamma: \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ ($a > b > 0$) has top vertex $C(0, 1)$, and the circle centered at $C$ with radius $\dfrac{4\sqrt3}{3}$ is tangent to $\Gamma$.

1. Find the equation of $\Gamma$. 2. A point $P$ on the circle sends two tangents to $\Gamma$, touching at $A$ and $B$. If $PA \perp PB$, find the coordinates of $P$.