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 $A.$

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