BAPC25-082.Boggle Sort

It is the 25th of February, 2025. You have enjoyed another spirited evening of Boggle with your friends. After everybody left, you have thoroughly cleaned the apartment. All that is left is to bring the Boggle tray in order. You start to wonder: would it be possible to bring the Boggle tray in alphabetic order, without swapping any dice, but only by rotating them?

The Boggle tray consists of $16$ six-sided dice. Each die is labelled with a letter from the English alphabet on each face. A single die contains a face labelled “Qu”. No letter appears $4$ or more times on the same die. By turning a die once, you can move any of the sideways-facing letters up. Turning a die twice moves the downwards-facing letter up.

Bring the tray into alphabetically nondecreasing order, using standard reading directions (left-to-right, top-to-bottom), using as few turns as possible. Letter case plays no role and the two-letter face is treated as “Q” followed by “U”, so “QuU” is sorted but “QuT” is not.