X divides the item into three pieces, let's call them I, II, and III, each of which in the opinion of X is worth exactly one-third of the total. X will end up getting one of these, yet we don't know which one. This knowledge forces X to make sure that the three pieces are worth one-third each. Otherwise the smallest piece might be the one left for her to take.
Both Y and Z declare or approve piece(s) they think are of size at least one-third. They normally have to do this simultaneously so that one doesn't know ahead of time the decision of the other. We can say here that the choosers (Y and Z) are declaring their bids. (Except in this case they are not bidding with money). A bider can approve one, two or all three of the pieces.
This is the case where we decide who gets what. Of course this will depend on who bids for which piece(s). There are two cases to consider:
In this case give Y and Z one piece each from the approved pieces and let X take the remaining piece.
Here we are momentarily in trouble. Why? Because both choosers (Y and Z) are interested in the same piece.
Well first is first; give X one of the pieces that are not approved. That should be "ok" because she was the divider and in her opinion any of the left pieces are worth exactly one-third.
Let's say piece III was the only approved piece and piece I it given to X. Then combine pieces II and III and divide the combined share, which in the opinion of Y and Z is worth at leat two-thirds, using the Divider-Chooser method.
Let's note that this method is proportional but not necessarily envy-free. Because X might envy whosoever ends up getting (in X's opinion) more than one-third after the second dividing and choosing.