Question: Let G be a group acting on a set X . Then there is an induced ( diagonal ) action of G over X

Let G be a group acting on a set X. Then there is an induced (diagonal) action of G over X \times X defined by g(x,y)=(gx,gy). We say that the action of G on X is doubly transitive if the induced diagonal action has exactly two orbits.
Show that if the action of Stab(y) is transitive for every y in X then the action of G on X is doubly transitive.

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