Simple question, but I don't think I've ever seen a discussion.  Let's say an opening bid of 1♠, a sound and unlimited 2♣ overcall, and responder cue bids 3♣ showing a limit raise or better of ♠.  What should advancer's double be?  Some considerations are: 1) if everyone has their bids, advancer won't have a strong hand, and 2) if advancer passes, the auction could well come back at 4♠.  Discussion of why one or another option is best would be much appreciated.

[Edit: added "lead directing" to choices as suggested]

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