\(\mathfrak{Square}^{\mathfrak{Square}}\)

Level pending

The squares 1, 25, 49 are pairwise relatively prime and in arithmetic progression. Let \(x_1^2, y_1^2, z_1^2\) and \(x_2^2, y_2^2, z_2^2\) be the next two smallest triples with this property when value is measured by the \(z_n\)'s. Find \(z_1+z_2\).

Note: This problem is not original.

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