
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
Four numbers, $c_1$, $c_2$, $c_3$, $c_4$, are drawn randomly and without replacement from the set $\{1, 2, 3, \dots, 500\}$. Assume that $c_1$ is always a prime number. Three other numbers, $d_1$, $d_2$, $d_3$, are then drawn randomly and without replacement from the remaining set of 496 numbers. Let $q$ be the probability that, after a suitable rotation, a brick of dimensions $c_2 \times c_3 \times c_4$ can be enclosed in a box of dimensions $d_1 \times d_2 \times d_3$, with the sides of the brick parallel to the sides of the box. If $q$ 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}
user