Flooring Tile - Slate Shower Surround

I have a shower surround that is slate tile. There is a small crack in the grout along one of the tiles. Also, the caulk is separating from the tub in one corner. I plan on re-caulking, and I think I have that down. Cut out the old caulk, clean everything real well and destroy any mildew, let it dry for a few hours, apply caulk, then go over it with a moist finger.

What I don't know is what to do about this crack in the grout. Should I regrout that section by cutting out all the grout and putting new grout in? Other sites seem to think that this is no big deal, but they are all referencing ceramic tile that is closely spaced. My slate tiles are about 1/4" apart. Do I therefore need a special grout? Could I just cover the crack with some caulk? Any advice would be appreciated.

Thanks in advance.

dig out any loose grout between the slate and regrout with sanded grout.(sanded grout for all joints over 1/8 " )
Good luck matching the color.
Slate and grout should be sealed.

Caulk in corners and at tub.