Suppose we have a "black$-$box" which is a program such that it can report either TRUE or

FALSE with any input number $x$ in constant time. Further, we have the following basic rules:

If the input $x$ is reported TRUE by the "black$-$box", then any number larger than $x$ must be

reported TRUE.

Given a sorted array $A[1,$dots,$n]$ of $n$ numbers, we want to find the smallest number in $A$

which is reported TRUE by the "black$-$box".

a$)$ Design an $O(logn)$ time algorithm for the problem. $(20)$

b$)$ Suppose the "black$-$box" needs $O(logn)$ time to report either TRUE or FALSE

instead of constant time, re$-$analyze the running time of your algorithm. $(10)$