
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 \( Q(x) = 2x^2 - 4x - 4 \). A real number \( x \) is chosen at random from the interval \( 4 \le x \le 12 \). The probability that \( \lceil \sqrt[3]{Q(x)} \rceil = \sqrt[3]{Q(\lceil x \rceil)} \) is equal to \( \frac{\sqrt{m} + \sqrt{n} + \sqrt{o} - p}{q} \), where \( m \), \( n \), \( o \), \( p \), and \( q \) are positive integers. Find \( m + n + o + p + q \).

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