Identify angle relationships to find x.

Since line AB is parallel to line CD, the angle $\angle FMA$ and the angle $\angle MNE$ are corresponding angles. Therefore, they are equal.

$$\angle MNE = \angle FMA = 5x$$

Angles $\angle MNE$ and $\angle CNE$ form a linear pair on the straight line CD. Their sum is $180^\circ$.

$$\angle MNE + \angle CNE = 180^\circ$$

Substitute $\angle MNE = 5x$ and $\angle CNE = x+78^\circ$ into the equation from Step 1:

$$5x + (x+78^\circ) = 180^\circ$$ $$6x + 78^\circ = 180^\circ$$

Subtract $78^\circ$ from both sides:

$$6x = 180^\circ - 78^\circ$$ $$6x = 102^\circ$$

Divide by 6:

$$x = \frac{102^\circ}{6}$$ $$x = 17^\circ$$

Since line AB is parallel to line CD, the angle $\angle AMN$ and the angle $\angle DNM$ are alternate interior angles. Therefore, they are equal.

$$\angle AMN = \angle DNM$$

Angles $\angle CNE$ and $\angle DNM$ are vertically opposite angles. Therefore, they are equal.

$$\angle DNM = \angle CNE = x+78^\circ$$

From these relationships, we can write:

$$\angle AMN = x+78^\circ$$

We are given $\angle AMN = 4x+y$. Equate this to the expression from Step 3:

$$4x+y = x+78^\circ$$

Substitute the value of $