
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 \(Q(x) = x^2 - 4x - 16\). A real number \(x\) is chosen at random from the interval \(4 \le x \le 20\). 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}
user