Three car accident and insurance says I'm 100% at fault even though the other car was tailing at high speed and hit first

As the title states, I'm in California and was involved in a 3 car accident where I was driving on a highway during commute time. Insurance says I'm 100% at fault and can appeal, but I'm wondering if I should seek out further legal advice before putting anything into writing?

While driving during commute time, the car in front of me suddenly hit the brakes and I saw it collide into the car in front of it (1st car). After seeing this, I applied brakes. I was about 2 car's length away when I applied brakes, but we were going at the speed of about 45-50mph. My car slowed down (3rd car), but could not entirely stop, coming into contact with the car in front of me (2nd car).

My car had damage to the bumper, which was on the minor side and the 1st car had damage to it's rear from the first impact. The center car (2nd car) had the most damage and could not even drive to the emergency lane. The front was smashed in and the rear had some minor dents and scratches. I did not push the 2nd car into the 1st car when I came into contact with the 2nd car. I mostly slowed down but did hit the 2nd car.

I pulled to the emergency lane where the 1st car was already waiting. The 2nd car, was disabled and could not drive off of the freeway. I spoke with the driver of the 1st car and he claims the lady (2nd car) was tailing and driving "way to close, she was way too close" and hit him. When highway patrol got onto the scene, we opted to file a police report and the officer interviewed us all separately. I was only able to get pictures and info from my car and the 1st car, since the 2nd car was stuck on the highway and pushed to another area away from us once police arrived. I could not get any evidence from the 2nd car who caused the accident.

I got my car fixed, but a letter in the mail from my insurance states that I am 100% at fault for the accident?! Which I think is absurd because technically there were 2 separate accidents. The 2nd car hitting the 1st car and then my car hitting the 2nd car (and not involving the 1st car at all). I think I should not be liable for any damage to the 1st car at all and not 100% at fault. I feel like the insurance did not look into the matter, otherwise they would seek some responsibility from the 2nd car?

The police report states that due to the 2nd car speed she was at fault and it also states the same for me. Insurance states I'm at fault.

The insurance letter states that I can file an appeal, but before doing that I wanted advice if I should just file the appeal or should I seek out legal advice before putting pen to paper again to insurance?