
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
Given the polynomial $Q(x) = x^2 - 5x - 15$, a real number $x$ is chosen at random from the interval $10 \le x \le 20$. Determine the probability that $\lfloor \sqrt{Q(x)} \rfloor = \sqrt{Q(\lfloor x \rfloor)}$. Express the probability in the form $\frac{\sqrt{f} + \sqrt{g} + \sqrt{h} - i}{j}$, where $f$, $g$, $h$, $i$, and $j$ are positive integers. Find $f + g + h + i + j$.

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