The Great Ant Walk

An ant is standing at the point \((0,0)\) in the Cartesian plane. The ant begins by walking \(a\) units in the negative \(y\) direction, for some positive integer \(a\). The ant then turns left and walks \(b\) units in the positive \(x\) direction for some positive integer \(b\). The ant again turns left and walks \(c\) units in the positive \(y\) direction for some positive integer \(c\). Finally the ant turns left once again and walks \(d\) units in the negative \(x\) direction, for some positive integer \(d\). An observer is watching the ant as it starts walking, but quickly gets bored of watching the ant. The observer only watched the first \(14\) units of the ant's walk, and noticed that the ant did not visit the same point on the plane more than once in this time. How many different possible paths could the observer have seen?

Details and assumptions

If the observer only watched the first 4 units of the ant's walk, then \( a = b = c = d = 1 \) is not a valid walk since the origin would be visited twice, namely at the start and at the end.

The ant walked for more than 14 units.

If the ant walked for \( a= 14 \), then all that the observer saw was the ant walking \(14\) units in the negative \(y\) direction. This is one of the possible paths.