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Buy a new tire.
That cannot be done. If broken, the tire is useless.
No, rotating the tires will not fix feathering. Tire feathering is an issue that must be fix, other wise you will keep on buy tires every 2 months ( if brand new), if use tire with in a week or so depending on the condition of the use tire. I learned this from personal experience.
Put an inner tube in the tire.
yes but you have to have a kit or at least a tire patch
Fix-a-Flat kits work on a very temporary basis. A punctured tire should be replaced.
yes
you don't. you replace it.
Yes.
Buy a tire pluging kit read the instructions and follow them
America's Tire Co and Discount Tire Co say that they will fix a flat tire for free they will even rotate all four tires for free. As far as I know there is no catch, they have always done this.
You cannot drive very fast with a fix-a-flat sealant in the tire. The exact speed depends on the weight of the car.