
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
Three numbers, \(a_1, a_2, a_3\), are drawn randomly and without replacement from the set \(\{1, 2, 3, \ldots, 500\}\). Three other numbers, \(b_1, b_2, b_3\), are then drawn randomly and without replacement from the remaining set of 497 numbers. Let \(p\) be the probability that, after any permutation, a brick of dimensions \(a_1 \times a_2 \times a_3\) can be enclosed in a box of dimensions \(b_1 \times b_2 \times b_3\), with the sides of the brick parallel to the sides of the box. Determine \(p\) as a fraction in the lowest terms and find the sum of the numerator and denominator.

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