2008 AIME II Problem 10

The diagram below shows a 4\times4 rectangular array of points, each of which is 1 unit away from its nearest neighbors.

Define a *growing path* to be a sequence of distinct points of the array with the property that the distance between consecutive points of the sequence is strictly increasing. Let m be the maximum possible number of points in a growing path, and let r be the number of growing paths consisting of exactly m points. Find mr.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.