
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 $a = \frac{p}{q}$ where $p$ and $q$ are positive integers that are relatively prime, satisfying the condition that the sum of all real numbers $x$ that fulfill
\[
\lfloor x \rfloor \cdot \{x\} = a \cdot x^2
\]
equals $272$. What is $p+q$?
A) 125
B) 136
C) 145
D) 153
E) 162

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