Squares and Reciprocals

We call a positive integer NN square-partitionable if it can be partitioned into squares of two or more distinct positive integers, and if the sum of the reciprocals of these integers is 1. For example, 4949 is square-partitionable because 49=22+32+62 and 12+13+16=1.49 = 2^2 + 3^2 + 6^2\quad \text{ and }\quad \frac{1}{2} + \frac{1}{3} + \frac{1}{6} = 1. What is the next smallest square-partitionable integer?

Note: This problem is intended to be solved with programming.

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