
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
A binary string is a word containing only  $0$ s and  $1$ s. In a binary string, a  $1-$ run is a non extendable substring containing only  $1$ s. Given a positive integer  $n$ , let  $B(n)$  be the number of  $1-$ runs in the binary representation of  $n$ . For example,  $B(107)=3$  since  $107$  in binary is  $1101011$  which has exactly three  $1-$ runs. What is the following expression equal to?   $$ B(1)+B(2)+B(3)+ \dots + B(255) $$  

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