
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  $ S(n) $  be the sum of the squares of the positive integers less than and coprime to  $ n $ . For example,  $ S(5) = 1^2 + 2^2 + 3^2 + 4^2 $ , but  $ S(4) = 1^2 + 3^2 $ . Let  $ p = 2^7 - 1 = 127 $  and  $ q = 2^5 - 1 = 31 $  be primes. The quantity  $ S(pq) $  can be written in the form   $$  \frac{p^2q^2}{6}\left(a - \frac{b}{c} \right)  $$   where  $ a $ ,  $ b $ , and  $ c $  are positive integers, with  $ b $  and  $ c $  coprime and  $ b < c $ . Find  $ a $ .

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