
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 \( \{1, 2, 3, \dots, 500\} \). 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 \( p \) 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 \( p \) 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