
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
Five numbers, \( a_1, a_2, a_3, a_4 \), are drawn randomly and without replacement from the set \( \{1, 2, 3, \dots, 500\} \). Four other numbers, \( b_1, b_2, b_3, b_4 \), are then drawn randomly and without replacement from the remaining set of 496 numbers. Let \( q \) be the probability that, after a suitable arrangement, a four-dimensional "hyperbrick" with dimensions \( a_1 \times a_2 \times a_3 \times a_4 \) can be enclosed in a "hyperbox" of dimensions \( b_1 \times b_2 \times b_3 \times b_4 \), with the sides of the "hyperbrick" parallel to the sides of the "hyperbox". 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}
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