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Dice mechanic with a given distribution?

I'm looking for a dice mechanic that produces a bell curve at all skill levels. I also want it to increase accuracy and precision as a character's skill increases - this basically means better average and smaller standard deviation. The simpler the system is the better, of course.

Skill + 2d6 increases accuracy as attribute/skill increases but precision remains the same.

Count 4+ in (Skill)d6 increases accuracy with skill but precision decreases as skill increases.

(Attribute)d(Skill) keep 2 where skill is 12, 10, 8, 6, 4 increases accuracy and precision as skill improves (gets smaller) but means there are only 5 steps for a given skill (or possibly attribute). It's also a system that requires a bunch of dice in each size. And has the "issue" of lower rolls being better. And that an increase in skill means a smaller dice while an increase in attribute means more dice.

(3 + Skill)d6 keep 3 increases both accuracy and precision but stops being a bell curve around 5 or 6 dice so provides few gradations in skill.

Skill + d20 is not a bell curve at all.

What other dice mechanics am I missing that might work for this system?