
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 $\{1, 2, 3, \dots, 500\}\,$. Another three numbers, $b_1\,$, $b_2\,$, $b_3\,$, are then drawn randomly and without replacement from the remaining numbers where the sum of $b_1 + b_2 + b_3\,$ must be greater than the sum of $a_1 + a_2 + a_3$. Determine the probability $p\,$ that a brick of dimensions $a_1 \times a_2 \times a_3\,$ will fit into a box of dimensions $b_1 \times b_2 \times b_3\,$ after including the constraint on sums. If $p$ is in reduced form, what is the sum of the numerator and denominator?

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