
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
5. Let  $S$  denote the set of all positive integers whose prime factors are elements of  $\{2,3,5,7,11\}$ . (We include 1 in the set  $S$ .) If  $$ \sum_{q \in S} \frac{\varphi(q)}{q^{2}} $$  can be written as  $\frac{a}{b}$  for relatively prime positive integers  $a$  and  $b$ , find  $a+b$ . (Here  $\varphi$  denotes Euler's totient function.)

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