Bob chooses a $4$-digit binary string uniformly at random, and examines an infinite sequence of uniformly and independently random binary bits. If $N$ is the least number of bits Bob has to examine in order to find his chosen string, then find the expected value of $N$. For example, if Bob’s string is $0000$ and the stream of bits begins $101000001 \dots$, then $N = 7$.

Let's think step by step and output the final answer within \boxed{}.
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