New years radical!!(in advance)

Algebra Level 5

Find the sum of all positive integers \((a,b)\) such that

\[\small{\small{\sqrt{\dfrac{(-a+b+\sqrt{a^2+b^2})(a-b+\sqrt{a^2+b^2})(a+b-\sqrt{a^2+b^2})}{4}\sqrt{\dfrac{(-a+b+\sqrt{a^2+b^2})(a-b+\sqrt{a^2+b^2})(a+b-\sqrt{a^2+b^2})}{4}\sqrt{\dots\dots}}}=4(a+b+\sqrt{a^2+b^2})}}.\]

Details and Assumptions

  • \((a,b)\) is same as \((b,a)\), no need to add twice.
  • This is inspired by Mursalin Habib.
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