I tried to make it less confusing.

ALL stabilizer is CYA--Cyanuric or Isocyanuric Acid. Easiest way to buy it is as a dry powder. There is a liquid version that works faster but is very, very expensive. The other two ways is from ordinary Di-Chlor powder and Tri-Chlor tablets.

You can use bleach FOREVER. I tend to buy liquid chlorine but only at one particular pool store because it tests out at 14% (though listed as 12%) and when compared to bleach is cheaper--as if I was buying 7% bleach at $1.50/gal. If its unit price goes higher than bleach, I'll use bleach instead.

While BBB stands for "Bleach, Borax and Baking Soda" what it REALLY means is you don't need pricey packaging from the pool store when the same stuff is far cheaper at the supermarket or big box store.