A disk of radius $2$ rolls all the way around the inside of a square of side length $s > 8$ and sweeps out a region of area $A_{1}$ . A second disk of radius $2$ rolls all the way around the outside of the same square and sweeps out a region of area $A_{2}$ . If $A_{1} = 16 s - 32 + 4 π$ and $A_{2} = 16 s + 8 π$ , and $A_{1} = \frac{2}{3} A_{2}$ , the value of $s$ can be written as $a + \frac{bπ}{c}$ , where $a, b,$ and $c$ are positive integers and $b$ and $c$ are relatively prime. What is $a + b + c$ ?