A Little Bit Of Poker Math

In Mathematics of Poker, Chen and Ankenman give the solution to the AKQ game.

The structure is: 2 players, X and Y each get one card from a deck containing 3 cards.  Betting is limit-betting, and the pot is P bets.  X checks, and cannot raise.

The solution given is:

Y bets all Aces, and 1/(P+1) of his Queens.  He always checks with a King.

Y calls all Aces, and (P-1)/(P+1) of his Kings.  He always folds a queen.

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So, that's "by the book".  What happens if you add a Jack to the deck?

Well, betting and calling with Aces remains the same.

Not betting and calling frequencies with Kings remains the same.

Not calling with Jack, and bluffing is the same.

But what happens with a Queen?  Classic weak bluff-catcher.  What do you do with it?