
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
Four numbers, \(a_1\), \(a_2\), \(a_3\), \(a_4\), are drawn randomly and without replacement from the set \(\{1, 2, 3, \dots, 1000\}\). Four other numbers, \(b_1\), \(b_2\), \(b_3\), \(b_4\), are then drawn randomly and without replacement from the remaining set of 996 numbers. Let \(q\) be the probability that, after a suitable rotation, a brick of dimensions \(a_1 \times a_2 \times a_3 \times a_4\) can be enclosed in a box of dimensions \(b_1 \times b_2 \times b_3 \times b_4\), with the sides of the brick parallel to the sides of the box. If \(q\) is written as a fraction in lowest terms, what is the sum of the numerator and denominator?

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