Each time it cuts out, has it run for a similar length of time?

If it's random, it may well be wiring related. However, by the sound of what you describe, and the fact that resetting the ballast (turning engine off and on again), it is more than likely an internal fault in the ballast causing it to trip out.

Have you tried swapping the main beam ballast for the dip beam ballast? Start with this (after checking the wiring). Doing the swap should show up whether it is a ballast fault or not, and give you a good idea whether the lamp is at fault too.

Where is your ballast mounted? If they get too hot they can cut out prematurely.

Yep... unfortunately.

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