Big sum of squares

\[\large 4\:475\:137 = 2016^2+641^2 \\ \large 4\:627\:633 = 1977^2+848^2\] The two large numbers above are prime numbers. Consider their product, \[N = 4\:475\:137 \times 4\:627\:633 = 20\:709\:291\:660\:721\]

Consider all possible ways in which \(N\) can be written as the sum of two squares: \[N = a^2 + b^2,\ \ \ a \geq b \geq 1.\]

What is the average value of the possible values of \(a\)? If you think \(N\) cannot be written as the sum of two squares, type 999.

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