AMC 12 2020a Problem 25

The number $a = \tfrac{p}{q}$, 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 $420$, 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 $x$. What is $p + q?$

$\textbf{(A) } 245 \textbf{(B) } 593 \textbf{(C) } 929 \textbf{(D) } 1331 \textbf{(E) } 1332$