Cassidy has string of $n$ bits, where $n$ is a positive integer, which initially are all $0$s or $1$s. Every second, Cassidy may choose to do one of two things:

1. Change the first bit (so the first bit changes from a $0$ to a $1$, or vice versa)
2. Change the first bit after the first $1$.

Let $M$ be the minimum number of such moves it takes to get from $1\dots 1$ to $0 \dots 0$ (both of length $12$), and $N$ the number of starting sequences with $12$ bits that Cassidy can turn into all $0$s. Find $M + N$.

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