
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
Let $B$ be the set of positive integers that have no prime factors other than $2$ and $7$. The infinite sum 
\[
\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{7} + \frac{1}{8} + \frac{1}{14} + \frac{1}{16} + \frac{1}{28} + \frac{1}{32} + \frac{1}{49} + \frac{1}{56} + \frac{1}{64} + \frac{1}{112} + \frac{1}{128} + \cdots
\]
of the reciprocals of the elements of $B$ can be expressed as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q$?
A) 8
B) 9
C) 10
D) 11
E) 12

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