
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 $b=\frac{\tilde{p}}{\tilde{q}}$, where $\tilde{p}$ and $\tilde{q}$ are positive integers without common divisors, has the characteristic that the sum of all real numbers $x$ satisfying
\[\lfloor x \rfloor \cdot \{x\} = b \cdot x^2 + \frac{1}{\lfloor x \rfloor}\]
is $4032$, 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 sum $\tilde{p} + \tilde{q}$.
A) 15
B) 929
C) 1260
D) 1331
E) 4032

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