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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