It is satisfied. We have cross-checked at that time and also answered your query.

Sum of Squares theorem states that a number $$$N$$$ can always be represented as the sum of two squares as long as there is no prime $$$p$$$ in $$$N$$$'s factorization such that $$$p \equiv 3$$$ $$$(mod$$$ $$$4)$$$ and its exponent is odd.

Because of the constraint on $$$N$$$, notice that it means that $$$N$$$ always takes the form of $$$x*y$$$, where $$$x$$$ is a power of $$$2$$$ and $$$y$$$ is less than or equal to $$$10^5$$$. Due to the Sum of Squares Theorem mentioned earlier, if $$$y$$$ is possible, then $$$x*y$$$ is possible, and if $$$y$$$ is not possible, then $$$x*y$$$ is also impossible. So it would be nice if we could find the answer for $$$y$$$, and translate that into an answer for $$$x*y$$$. Turns out that we can, since notice that every other power of 2 is a square. If we ensure that $$$x$$$ is a power of $$$4$$$(and thus, a perfect square), $$$y \leq 2*10^5$$$ will hold true.

As such, we can precalculate the answer for everything from $$$1...2*10^5$$$, and when answering a query of $$$N$$$, we split $$$N$$$ into $$$x*y$$$. Our answer should then be $$$\sqrt{x}*(y_1,y_2)$$$, where $$$y_1$$$ and $$$y_2$$$ are the valid pair(if it exists) for $$$y$$$.