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Multiplicative decomposability of shifted sets

The following two problems are open. Do two sets of positive integers 𝒜 and ℬ exist, with at least two elements each, such that 𝒜+ℬ coincides with the set of primes 𝒫 for sufficiently large elements? Let 𝒜={6, 12, 18}. Is there an infinite set ℬ of positive integers such that 𝒜ℬ+1⊂𝒫? A positive answer would imply that there are infinitely many Carmichael numbers with three prime factors.