In a rectangular plot of land, a man walks in a very peculiar fashion. Labeling the corners A B C D, he starts at A and walks to C. Then, he walks to the midpoint of side A D, say A_{1}. Then, he walks to the midpoint of side C D say C_{1}, and then the midpoint of A_{1} D which is A_{2}. He continues in this fashion, indefinitely. The total length of his path if A B=5 and B C=12 is of the form a+b \sqrt{c}. Find \frac{a b c}{4}.