
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
Consider the number $a = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. This number has the property that the sum of all real numbers $x$ satisfying the equation
\[\lfloor x \rfloor \cdot \{x\} = a \cdot x^2\]
is $360$, 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 $p+q$.
A) 245
B) 593
C) 701
D) 929
E) 1331

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