
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
The number \(a=\frac{p}{q}\), where \(p\) and \(q\) are relatively prime positive integers, has the property that the sum of all real numbers \(x\) satisfying \(\lfloor x \rfloor \cdot \{x\} = a \cdot x^2\) is \(504\), where \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \(x\) and \(\{x\}=x- \lfloor x \rfloor\) denotes the fractional part of \(x\). Determine the value of \(p+q\).

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