
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 $2$, $3$, $5$, and $7$. Consider the infinite sum of the reciprocals of the elements of $B$:
\[
\sum \frac{1}{n} \quad \text{for} \quad n \in B
\]
where each $n$ can be expressed as $2^a 3^b 5^c 7^d$ with $a$, $b$, $c$, and $d$ as non-negative integers. This sum can be expressed as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q$?
A) 36
B) 40
C) 42
D) 43
E) 45

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