It seems to me that the other example Command spell uses all at least strongly imply "do nothing but this". Approach doesn't specifically state that the target has to use their action (say, Dash) to approach you, just that they approach you, and only specifies ending the turn at that point if they get within 5 feet.
Option 1: Implicitly, they must take the Dash action if they aren't yet within 5 feet.
Option 2: They don't have to take Dash, but can't do other things.
Option 3: They move. If this brings them within 5 feet, they end their turn, otherwise they can now use their action and/or bonus action normally.
I'm sort of inclined to Option 2, but I think 1 might also make sense, because "follow the command on its next turn" appears to imply "do nothing on the next turn but follow the command".
Option 1: Implicitly, they must take the Dash action if they aren't yet within 5 feet.
Option 2: They don't have to take Dash, but can't do other things.
Option 3: They move. If this brings them within 5 feet, they end their turn, otherwise they can now use their action and/or bonus action normally.
I'm sort of inclined to Option 2, but I think 1 might also make sense, because "follow the command on its next turn" appears to imply "do nothing on the next turn but follow the command".