
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, $c_1\,$, $c_2\,$, $d_1\,$, $d_2\,$, are drawn randomly and without replacement from the set $\{1, 2, 3, \dots, 100\}\,$, such that the sum of $c_1$ and $c_2$ is always greater than 150. Assume that $c_1 \geq c_2$ and $d_1 \geq d_2$. What is the probability $q$ that a rectangle of dimensions $c_1 \times c_2$ can be enclosed in a rectangle of dimensions $d_1 \times d_2$, with the sides of the rectangle parallel to each other? 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}
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