Imagine a game in which you have the integers from 1 to 12 written on slips of paper. You roll 2 dice and take their sum. You can then flip over any combination of pieces of paper with a sum equal to the sum of the dice. (For example, if you roll a 5 and a 2, you could flip over the 7; the 6 and 1; the 5 and 2; the 4 and 3; or the 4, 2, and 1.) Once a piece of paper is flipped over, you cannot use that number anymore. You win if you flip over all 12 pieces of paper.
Two questions:

What is the best strategy? (I believe you should always avoid flipping small numbers and try to flip the single number equal to the sum you rolled, if possible. I base this on the fact that if you roll a 12, you’d be foolish to flip anything other than the 12, and so on.)

Is there an easy way to calculate the probability of winning the game? (This seems to me to be a situation with way too many branches to really evaluate, but maybe there’s some trick to think about it from the right perspective.)
This isn’t a homework problem.