
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 $Q(x) = x^2 - 5x - 15$. A real number $x$ is selected randomly from the interval $7 \le x \le 17$. The probability that $\lfloor \sqrt{Q(x)} \rfloor = \sqrt{Q(\lfloor x \rfloor)}$ is $\frac{\sqrt{u} + \sqrt{v} + \sqrt{w} - z}{y}$, where $u$, $v$, $w$, $z$, and $y$ are positive integers. Find $u + v + w + z + y$.

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