Squares and Reciprocals

We call a positive integer \(N\) 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, \(49\) is square-partitionable because \[49 = 2^2 + 3^2 + 6^2\quad \text{ and }\quad \frac{1}{2} + \frac{1}{3} + \frac{1}{6} = 1.\] What is the next smallest square-partitionable integer?

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

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