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An imprecise auction

Playing with Annie on BBO against good friends.

North
xx
KQ10
AKJ8xx
xx
South
AK108xx
AJ7xxx
A
W
N
E
S
1
P
2
P
2
P
3
P
3
P
3
P
?

We play precision but, after the first round, this sequence is a very natural one. 1C was 16+hcp, 2D was 12+hcp with 5+ diamonds. Just because many people like polls, I will ask what you think should be South's next two bids. But I think there is more than just a bidding poll here and I am hoping to get thoughts on other aspects of the deal.

South bid 4C, West doubled, North passed denying help in clubs, East passed and South redoubled to confirm a club control. North's pass was good news in that it meant they did not have wasted club values.

After the redouble, North bid 4D and now South had a real problem. 4H would be a control bid. 4N would be keycard in spades. I don't know for sure what 5H would be, but I would think it should be a spade slam invite asking North if they had help in hearts,  5S would also be an invite focusing on general strength and trump in context, 5N would be "pick a slam" but we do not have a rule that says, "6N is not one of your choices". (Should we?) And I guess 6H is natural although in other auctions "one under" is last train.

Where do you want to be? It would appear to be 7H which requires diamonds 4-3, or 5-2 with the DQ doubleton (call that 63+2/7x30=71%) but that requires hearts not 4-0. Alternatively, you make when spades are 3-2 (68%) and you can combine your chances by winning the opening lead of the CK and crossing to dummy in hearts. Even though the two are not independent, I make the grand as being easily over 80%. (Note you have 13 tricks if a clever opponent tries the ploy of playing the DQ on the first round of the suit.)

Our friend pointed out that our precision auction preempted us. (Other than relay, I think club systems are all going to suffer a similar fate. For example, I believe R-M would also find opener showing hearts for the first time at the 3 level.) Standard bidders would have bid 1S-2D-2H-3D-3H-4H. The auction isn't over but at least N-S have found the right trump suit. At this point, one possibility is keycard in hearts which works when North shows 2 plus the HQ. But I entertain suggestions for other ways of getting to 7H. 

So, what is your plan as South after North's 3S bid? If you choose 4C, assume it goes (DBL)-P-(P)-RDB-(P)-4D-(P) back to South.

4C, then 4H
4C, then 4S
4C, then 4N
4C, then 5H
4C, then 5N (pick a slam)
4C, then 6H
4H, then hope for a second bid
4N, keycard in spades and upon discovering the SQ missing, 6H as an alternative trump suit
5H
5N
6H
Other

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