Alice and Bob are playing a guessing game. Bob is thinking of a number n of the form 2^{a} 3^{b}, where a and b are positive integers between 1 and 2020, inclusive. Each turn, Alice guess a number m, and Bob will tell her either \operatorname{gcd}(m, n) or \operatorname{lcm}(m, n) (letting her know that he is saying that gcd or lcm), as well as whether any of the respective powers match up in their prime factorization. In particular, if m=n, Bob will let Alice know this, and the game is over. Determine the smallest number k so that Alice is always able to find n within k guesses, regardless of Bob’s number or choice of revealing either the \mathrm{lcm}, or the gcd.