This is a continuation of MULTIPLE Choice (part 1).

Why we cannot allow an empty set as an answer? Why not, when we do have a symbol for empty set?

A problem has 4 answer choices (A, B, C and D). The correct answer is a subset of the set of all answer choices (it might be empty). 2 points will be given if your answer is exactly identical to the correct answer, and 1 point will be given if your answer is a nonempty subset of the correct answer (so submitting an empty set for a problem with nonempty answer subset still gives you nothing). However, if the answer is indeed an empty set and you get it right, you will get \(x\) bonus points (in addition to the 2 points), for being brave! (\(x\) is an integer, since teacher hates fractions in grading.)

Again you confronts such a problem. And again (oops) you are not brilliant enough to figure out anything. When you are trying to use the same strategy as last time, that is to maximize the expected number of points assuming all subsets have equal chance of being correct, you realized that submitting an empty set is in fact the best strategy now (strictly better than every other subset). Then what is the minimum value of \(x\)?

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