In a construction project, Joe decides to stick up \(5\) vertical poles, of lengths \(1\), \(3\), \(5\), \(7\), and \(9\). Each pair of consecutive poles have diagonal strands attached from the top of one pole to the bottom of the other. From the intersection of these diagonal strands, more poles are vertically stuck up. Using only the new poles, this process is repeated until one pole remains. To the nearest thousandth, find the length of this pole.
This problem is part of Tristan's set Formulas and Theorems.