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Yes, they would definitely do something.
The main relativity concept is that all physics rules are relative to your reference frame.
So, you are NOT traveling at the speed of light, but it's the rest of the universe that's traveling at the speed of light towards you.
If you switch on your headlight, photons would start traveling ahead of you as they would do "normally", that is when you are at rest with respect to the rest of the universe.
