reconstructed title - auto accident

Recently purchased, fully insured, and totaled a car with a reconstructed title. How do I get the insurance company either to pay the repairs or the amount I have owed on it?
Can they total it with even a window broken if the title says it has no value? How can it be registered and pass inspection with no value? The insurance agent said the title would'nt matter when I went to insure it fully. The car was totaled during a rainstorm when a white new pick up truck cut immediately in front of me and I had no recourse but to swerve to avoid a serious crash and hydroplaned into a telephone pole.I was not at fault due to the fact I was forced to avoid an accident caused by the truck. I was wearing my seat belt and was not hurt but just bought the car for $3000.00 and its retail value should be over $4000.00 and trade-in around $2500.00. I wouldn't mind having it repaired as I don't see it being totaled as they have no value to total it against.
Please advise asap.
Sincerely,
Michelle

If you are insuring it fully, that usually means that you are insured against damage to others (liability, which is required in most states) and damage caused to your own vehicle (split up into collision - accident damage, and comprehensive - damage caused by vandalism, hail, rocks on the road, etc.). It would make no sense to pay the collision portion of the insurance if the car has no insurable value. Something's wrong there. Are you sure you paid for collision coverage?