Conic Sections · Superchallenge problem

The hyperbola $\dfrac{x^2}{4} - \dfrac{y^2}{12} = 1$ has foci $F_1 = (-4, 0)$ and $F_2 = (4, 0)$. Let $A = (1, 4)$. As $P$ moves along the right branch of the hyperbola, find the smallest possible value of $PF_1 + PA$.