Who would be at fault in this car accident?

Earlier today i was driving around 30 mph and there was a car following very close so i decided to speed up cause that annoys me when they ride on your bumper. So i had enough distance, but he was still pretty close and a cat ran out in front of my car, and i had no time to think because i was going about 35 or 40 mph, so my first reaction was stomping on the brake pedal, and i began skidding, but the ABS limited the sliding. At first i thought i hit the cat cause i couldn't see it because it was under my car, but i saw the cat run away so i didn't hit it, but after i slamed on my brakes to avoid the cat, i saw the car that was behind me slam on his brakes and come really really close to rear-ending me. Now im an animal lover, and would feel real bad if i killed a cat, but at the same time, i don't want to total two cars and maybe hurt me and the driver because of a cat. So my question is, had the guy who was riding my bumper rear-ended me, would it be his fault for following to close and not paying attention or would it be mine for stopping so quick rather than swerve to miss it.