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Kokish after 2 openings?

I'm curious about the popularity of Kokish after a strong, artificial 2 opening bid.

I define Kokish as using a 2 rebid, after 2-2, to show either a GF(+) balanced hand, or a natural 2 bid.  Responder usually (or for some, always) bids 2, and opener completes the description.

Regardless of the version used for followups, let's assume that anything that starts this way is Kokish.

One thing I've noticed is that a number of pairs in my area seem to be using Kokish in conjunction with a direct 2 response as a double negative, and 2 GF.  The first time I heard this, I said "impossible", but I've seen it several times since, and have even agreed to play it with a couple of partners, since I often agree to play my partner's system.

My question is intended especially for those who have given at least some thought to their preferences in responding to 2, and have an opinion on what they prefer.  If you don't have a preference, perhaps because you don't have a lot of experience with these auctions, or tend to simply go along with your partner's preferred system, feel free to abstain without explanation in order to see the responses.  Thanks in advance!

Here goes my question:

When using 2 as a Strong artificial opening, which includes GF balanced hands, which of the following methods to you prefer to agree to?

Kokish after a GF 2 response, with a direct 2 response defined as a double negative
Kokish after a 2 response, with the 2 response defined as waiting, controls, or something similar
Kokish, but the 2 response has some other definition not covered by answer 1 or 2
A response structure which does not involve Kokish at all

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