
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
Five numbers, \( c_1, c_2, c_3, c_4, c_5 \), are drawn randomly and without replacement from the set of even numbers between 1 and 100, inclusive. Let \( p \) be the probability that, after a suitable rotation, a brick of dimensions \( c_1 \times c_2 \) can be enclosed in a box of dimensions \( c_3 \times c_4 \), with the sides of the brick parallel to the sides of the box. If \( c_5 \) is used to increase one dimension of either the brick or box by \( c_5 \) before trying to fit the brick in 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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