
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
An \textit{island} is a contiguous set of at least two equal digits. Let  $b(n)$  be the number of islands in the binary representation of  $n$ . For example,  $2020_{10} = 11111100100_2$ , so  $b(2020) = 3$ . Compute
\[b(1) + b(2) + \cdots + b(2^{2020}).\]

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