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Pass/bid after they double partner's control cue

Let's say we have an auction like the following:

W
N
E
S
1
P
2NT
P
3
P
4
X
?

It's fairly universal that if North passes, this allows South to redouble with first-round control. Likewise, North's redouble would show first-round control.

What about the situation where South has the King, but not the Queen, and is worried that AQ is sitting over them? 

Does North have some way to show that they have a singleton/Queen of diamonds, i.e. control of the second round of the suit counting partner's control? How do we reassure South that we aren't losing two quick tricks? What's the standard agreement here?

Likewise, if we have nothing in the suit, is there a way for South to show they hold KQ/singleton of the suit, rather than Kx/Kxx or similar?

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