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It's always good to keep track of them as you go. At least look at the numbers you're starting out with and make sure that the sig figs at the beginning of the problem match the end of the problem. However, as you're working along the problem, you can be off in terms of sig figs one or two places and generally that doesn't affect the final answer. Just be mindful of rounding off by large margins when there's multiple steps in a problem and you should be good.
I was always told to not round off any numbers until the very end. This could be especially important considering the small numbers discussed in chemistry. So my suggestion would be, when in doubt, work through the problem without condensing any of the numbers and then apply sig figs to your answer. Then make sure your answer has as many sig figs as the question.
When making an intermediate calculation in a problem with multiple steps/operations, I usually underline the digit of the last sig fig but leave the number unrounded. It helps me to keep track of the number of sig figs from step to step without losing accuracy by rounding during intermediate steps.
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