In the figure, [tex]ABCD[/tex] is a square whose side length is [tex]1[/tex]. Draw a quadrant of a circle with [tex]A[/tex] as the center and [tex] B, D[/tex] as the endpoints of the arc. Similarly, draw a quadrant of a circle with [tex]B[/tex] as the center and [tex] A, C[/tex] as the endpoints of the arc. A circle with center [tex]O[/tex] is also inscribed touching the arc [tex]AC[/tex] externally, arc [tex]BD[/tex] internally, and side [tex]AD[/tex]. If the diameter of the circle is [tex]x/y[/tex] for coprime positive integers [tex]x[/tex] and [tex]y[/tex], find [tex]x+y[/tex].​