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?

(第十一届AIME1993 第7题)