inequality – Minima of $x+y+frac{1}{xy}$ given $x^2+y^2=1$

If $x,y$ are positive reals satisfying $x^2+y^2=1$, then the minimum value of $x+y+frac{1}{xy}$ is?

Attempt 1

I used the Lagrange multipliers method which ended up being cumbersome.

Attempt 2

I applied the AM-GM inequality to obtain $frac{1}{xy}ge2$ and $x+ylesqrt2$ after which I’m lost again.

Im looking for hints/better approaches to the problem.