You have access to rand5(), which returns an integer in {1,2,3,4,5} uniformly at random. A) Implement rand7() that returns {1..7} uniformly using only calls to rand5(). Your solution must be correct (unbiased), terminating, and near-optimal in expected number of rand5() calls. Prove uniformity and compute the exact expected number of rand5() calls your construction uses. B) Generalize: given randM() (uniform over {1..M}), implement randN() (uniform over {1..N}) for arbitrary positive integers M and N. Describe an algorithm that is correct for all M,N, and derive an upper bound on the expected number of randM() calls in terms of M and N. Discuss how you would adapt when N≫M and when M≫N. C) Biased source: given toss(p) which returns 1 with unknown probability p∈(0,1) and 0 otherwise, construct fair_coin() that returns 0/1 with equal probability, prove correctness, and derive the expected number of tosses as a function of p. Then extend your approach to sample uniformly from {1..K} using only toss(p). Analyze termination and expected complexity.