Suppose that $\triangle ABC$ is an equilateral triangle of side length $s$, with the property that there is a unique point $P$ inside the triangle such that $AP = 1$, $BP = \sqrt{3}$, and $CP = 2$. What is $s?$ $\textbf{(A) } 1 + \sqrt{2} \qquad$ $ \textbf{(B) } \sqrt{7} \qquad$ $ \textbf{(C) } \frac{8}{3} \qquad$ $ \textbf{(D) } \sqrt{5 + \sqrt{5}} \qquad$ $ \textbf{(E) } 2\sqrt{2}$