The cheat button can be used to check states for you. It's notion of depth is a little different from moves, if a state only has one possible move it will follow forced options until reaching a state with options, even if all but one of them is known to be dead.
The first time you click the cheat button from a game state it checks each child state (the game state you get to after performing a move) If a child state only has one viable move it checks the state after that move and if that also only has one move it checks that and so on. The result is that on the first click of cheat button if a move leads to a totally forced sequence of moves that leads to a dead state it marks that move as dead, which you can see by using cheat from the initial state in this level, or after making one move of A.
After the first use of cheat subsequent uses analyse new states, all states at the next depth (corresponding to choices which is every move from a state with more than one valid move). It informs you how many states it visited which can help identify dead ends, if the number keeps dropping to 0 then it is likely a complicated dead end, but if it grows rapidly then there is probably a state down the line that frees up an empty tube.