
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 $Q(x) = x^2 - 4x - 4$. A real number $x$ is chosen at random from the interval $3 \le x \le 10$. The probability that $\lfloor\sqrt{Q(x)}\rfloor = \sqrt{Q(\lfloor x \rfloor)}$ is equal to $\frac{\sqrt{p} + \sqrt{q} - r}{s}$, where $p$, $q$, $r$, and $s$ are positive integers. Find $p + q + r + s$.

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