A walk on Muhammad's plot

Geometry Level 4

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.

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