Jumping Virginia Gentleman

A recent retiree, 58 yo, successful in business, decides to adopt a new sport, equestrian obstacle racing. Although generally a good athlete in his youth, the sedentary decades have taken their toll. Our retiree trains regularly and progresses to where he can often stay in his saddle. But, it is also not uncommon to see him fly off into space while attempting to leap obstacle, as the image shows. His expectant fifth wife is worried. After one such airborne episode, he reports to his examining physician that even at his desk, he suffers episodic headaches and is experiencing progressive weakness and paralysis on the left side of his body. When auscultating the neck, the physician hears the sound of a bruit on the left side of the anterior neck. Angiography of the affected vessel indicates 72% luminal narrowing. Our recent retiree is a good surgical candidate, so an endarterectomy is recommended. During this surgery an incision is made to access the vessel, the occlusion is removed and the vessel is left smooth. Which nerve travels with the affected artery and is at risk of damage during this procedure?