A walk on Muhammad's plot

A man owns a rectangular plot of land with corners labeled $ABCD$ , with $AB=5$ meters and $BC=12$ meters. He walks in a very peculiar fashion. He starts at $A$ and walks to $C$ . Then, he walks to the midpoint of side $AD$ (labeled $A_1$ ). Then, he walks to the midpoint of side $CD$ (labeled $C_1$ ), and then the midpoint of $A_1D$ (labeled $A_2$ ). He continues in this fashion indefinitely. The total length of his path is of the form $a + b\sqrt{c}$ , where $a,b$ and $c$ are positive integers, and $c$ is not divisible by the square of any prime. What is the value of $a+b+c$ ?

This problem is posed by Muhammad A.