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The worker stumbled, lost a ski pole and failed to leave the Dave Murray piste before Branch came hurtling towards the finish area.
"Obviously, it was super dangerous, not only for him but also for me," Branch told reporters. "If we had collided, there would have been a huge risk of a major injury."
The Australian, who could not change his line at that point, sped straight ahead to the finish without touching the worker.
"I landed on the last jump, I looked at the finish line and noticed that somebody was very close to the race line," Branch told reporters after narrowly avoiding a horrific crash.
"I couldn't really tell if he was moving or not. I just took my line to the finish, put my head down and hoped that he didn't move any further because I couldn't change my line.
"It's scary but it can happen," he added. "Luckily nothing serious happened. I'm sure they will look into it."
Branch finished 34th, nearly three seconds behind Swiss winner Didier Defago.
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