
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 $\{1, 2, 3, \dots, 1000\}\,$. Another four 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 each side of the brick parallel to a side of the box. If $q\,$ is expressed as a fraction in lowest terms, what is the sum of the numerator and denominator?

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