Auto Insurance Claims: ARE THERE TIME LIMITS THAT A INS CO MUST DECIDE LIBILITY FOR A CAR ACCIDENT, oncoming traffic lane, iwas

Question
I WAS INVOLVED IN A ACCIDENT WHEN DRIVING WHEN A CAR SUDDENLY BACKED OUT OF A DRIVEWAY.I BLEW MY HORN , SWERVED TO THE LEFT TO THE ONCOMING TRAFFIC LANE AND DRIVER BACKED OUT OF DRIVEWAY AND HE HIT THE FRONT RIGHT HAND SIDE OF MY FRONT BUMPER.IHAD 3 PASSENGERS IN MY CAR AT THE TIME WE EXCHANGED ALL INFORMATION ,I CALLED HIS INS CO AND GOT A CLAIM# AND HAD THERE ADJUSTER LOOK AT THE CAR.THE DAMAGE WAS A FEW $500 IWAS TOLD THAT A LIBILITY ADJUSTER WOULD CALL ME,I REPORTED IT TO MY INS CO BUT TOLD THEM I AM DEALING WITH THE OTHER INS CO AND DONT WANT THEM TO DO ANYTHING THEY SAID FROM THE INFO I GAVE THEM THE OTHER DRIVER WAS AT FAULT.THE OTHER INS CO SAID THAT THEY HAVENT HEARD FROM THE OTHER DRIVER.WHAT I WANT TO NO IS THERE A TIME LIMIT THAT I HAVE TO WAIT FOR THE OTHER INS CO TO MAKE A DECISION ON  LIBILITY OR SHOULD I HAVE MY INS CO HANDLE THE CLAIM BUT I WILL PAY FOR THE DAMAGE MYSELF AND WAIT FOR THEM TO RESOLVE IT WITH THE OTHER CO

Answer
Since auto insurance is regulated by the states and the time limits vary, your best choice is to file the claim with your own company and let them go after the other party's insurer. Once they collect from them in the subrogation process, your deductible will be refunded to you.