Suppose Alice and Bob are using a cryptosystem with a $128-$bit key, so there are $2^{128}$ possible keys. Eve is trying a brute$-$force attack on the system.

a$.$ Suppose it takes $1$ day for Eve to try $2^{64}$ possible keys. At this rate, how long will it take for Eve to try all $2^{128}$ keys? $($HINT: The answer is not $2$ days.$)$

b$.$ Suppose Alice waits $10$ years and then buys a computer that is $100$ times faster than the one she now owns $($so it takes only $1/100$ of a day, which is $864$ seconds, to try $2^{64}$ keys$).$ Will she finish trying all $2^{128}$ keys before or after what she does in part $($a$)?($NOTE: This is a case where Aesop's Fable about the Tortoise and the Hare has a different ending.$)$