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Checkback choices and continuations

In a Polish Club context with a relatively new partner you have the following auction:

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

1 = three-way Polish: (11)12-14 balanced or 4414, 15+ clubs or 18+ anything;

1 = natural, 4+ hearts, 7+ HCP or equivalent (GF opposite the strong 18+ option);

2N = 18+ balanced or semi-balanced, max 3 hearts and usually not 3 (GF now established).

With no previous discussion, I alerted 3 as there was a significant chance it wasn't natural. Based on system options elsewhere this is either simple Checkback or natural. The undiscussed meaning of 3 is likely to be the same, denying 3 hearts and showing 4 spades, but if 3 is strictly natural then having 3 as natural is not optimal. 

I have no issue with the auction in the context of undiscussed territory. 4 was an acceptable if not ideal contract (3N probably better at MP). I have various thoughts going forward, and would appreciate suggestions on:

  1. What do people think we *should* be playing after 2N here?
  2. If 3 is simple Checkback, what should 4 mean?
  3. What should 4/4 mean after 3?
  4. If 3 is natural, what should 3 mean?

 

Question 1 is the most interesting to me. Clearly 'standard' 2-way Checkback isn't helpful as we're already GF, but similarly a natural or simple Checkback approach can't be the optimal use of bids.

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