Where have you seen stabilizer called "clarifier" -- if someone in a pool store is calling it that, I'm pretty sure they are just confused.
The relationship between CYA and chlorine is more a constant fraction, rather than a constant PPM. That is, under ideal circumstances, if your chlorine ppm is 5% of your CYA ppm, everything's good. On the other end, sometimes killing mustard algae takes a chlorine ppm that's 25% or more of your CYA level. The relationships are not all mathematically linear, so that's not perfectly true. For example, a chlorine level of zero is not going to help, even if your CYA is zero.
But, it's approximately true.
Finally -- your pool doesn't care how the chlorine gets there: chlorine gas, bleach (sodium hypochlorite), dichlor (sodium dichloro-isocyanuriate) all work. But if you use store bleach, get the PLAIN store brand 8% bleach, and not the 'flavored' bleach that Clorox has been marketing.
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