
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
Consider sequences \( a \) of the form \( a = \left(a_{1}, a_{2}, \ldots, a_{20}\right) \) such that each term \( a_{i} \) is either 0 or 1. For each such sequence \( a \), we can produce a sequence \( b = \left(b_{1}, b_{2}, \ldots, b_{20}\right) \), where
\[ 
b_{i} = \begin{cases} 
a_{i} + a_{i+1} & \text{for } i = 1 \\
a_{i-1} + a_{i} + a_{i+1} & \text{for } 1 < i < 20 \\
a_{i-1} + a_{i} & \text{for } i = 20 
\end{cases}
\]
How many sequences \( b \) are there that can be produced by more than one distinct sequence \( a \)?

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