On the coordinate plane, for a natural number $a > 1$, consider the two curves $y = 4 ^ { x } , y = a ^ { - x + 4 }$ and the line $y = 1$. Find the number of values of $a$ such that the number of points with integer coordinates inside or on the boundary of the region enclosed by these curves and line is between 20 and 40 (inclusive). [4 points]