
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
Let $P(x) = x^3 - 4x^2 - x + 20$. A real number $x$ is chosen at random from the interval $3 \leq x \leq 10$. The probability that $\lfloor \sqrt[3]{P(x)} \rfloor = \sqrt[3]{P(\lfloor x \rfloor)}$ is equal to $\frac{\sqrt{a} + \sqrt{b} + \sqrt{c} - d}{e}$, where $a$, $b$, $c$, $d$, and $e$ are positive integers. Find $a + b+ c + d + e$.

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