Albert is taking a test where each of the $1000$ questions has one of two answers: true or false. Having not prepared for the test, he does not know the answer to a single problem and resorts to intelligent guessing. He knows that the number of questions on the test with answer true is a power of two, with each power of two having an equal probability and each possible distribution being equally likely for a certain true count. Assuming Albert guesses optimally, how many questions is he expected to answer correctly?