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 $A.$