
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
Given a known sequence of distinct real numbers \(r_{1}, r_{2}, r_{3}, \cdots, r_{n}\), performing a single operation involves comparing the second element with the first one, swapping them if the second one is smaller, and continuing this process up to the last element. The sequence \(1, 9, 8, 7\) demonstrates how this operation transforms it into \(1, 8, 7, 9\). Assume \(n=40\) and the sequence \(r_{1}, r_{2}, \cdots, r_{40}\) consists of distinct real numbers randomly arranged. Let \(\frac{p}{q}\) (in lowest terms) represent the probability that the original 20th element \(r_{20}\) moves to the 30th position after one operation. Find the value of \(p+q\).

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