
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
From the set \(\{1, 2, 3, \ldots, 1000\}\), three numbers \(a_{1}, a_{2}, a_{3}\) are randomly selected without replacement. Then, from the remaining set of 997 numbers, another three numbers \(b_{1}, b_{2}, b_{3}\) are randomly selected without replacement. Let \(p\) be the probability of the following event: A brick with dimensions \(a_{1} \times a_{2} \times a_{3}\) can be placed inside a box with dimensions \(b_{1} \times b_{2} \times b_{3}\), with the sides of the brick parallel to the sides of the box, after suitable rotations. If \(p\) is written as a reduced fraction, what is the sum of the numerator and the denominator?

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