the problem is keeping the bleach in the area that needs to be bleached. I have heard of people soaking bread slices in bleach and laying them over the intended area, but never tried it.
Whatever you do to clean the grout, it will continue to come back. You are experiencing one of the biggest drawbacks of a grouted tile counter top.
When new, before used, a silicon sealer would have helped to a degree, but now that is too late.
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