# 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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