
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 \( Q(x) = x^2 - 5x - 20 \). A real number \( x \) is chosen at random from the interval \( 0 \le x \le 20 \). The probability that \( \lfloor \sqrt{Q(x)} \rfloor = \sqrt{Q(\lfloor x \rfloor)} \) is equal to \( \frac{\sqrt{a} + \sqrt{b} - d}{e} \), where \( a \), \( b \), \( d \), and \( e \) are positive integers. Find \( a + b + d + e \).

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