
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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  $M + N$ .

Present the answer in LaTex format: \boxed{Your answer}
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