
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
Suppose \( p \) and \( q \) are two different positive integers and \( x \) is a real number. Form the product \( (x+p)(x+q) \).

(a) Find the sum \( S(x, n) = \sum (x+p)(x+q) \), where \( p \) and \( q \) take values from 1 to \( n \).

(b) Do there exist integer values of \( x \) for which \( S(x, n) = 0 \)?

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