
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  $\{1, 2, 3, \dots, 1000\}$ .  Three other numbers,  $b_1$ ,  $b_2$ ,  $b_3$ , are then drawn randomly and without replacement from the remaining set of 997 numbers.  Let  $p$  be the probability that, after a suitable rotation, 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.  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}
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