When Does This Point Not Exist?

Let \(\triangle ABC\) be an acute angles triangle with \(\angle BAC = 80^{\circ}.\) Let \(M\) be the midpoint of \(BC,\) and let \(O\) be the circumcenter of \(\triangle ABC.\) Suppose there exists no point \(X\) in the plane of \(\triangle ABC\) which satisfies the following conditions.

• \(X \neq O, X \neq A\)
• \(\angle BAX = \angle CAM\)
• \(\angle AXO = 90^{\circ}\)

Let \(\angle ABC = N^{\circ}.\) Find \(\dfrac{N}{2}.\)

Details and assumptions
- The diagram shows the point \(X,\) which shouldn't exist.
- The first condition means \(X\) must be different from \(O\) and \(A.\)