
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
Dave arrives at an airport which has twenty gates arranged in a straight line with exactly $50$ feet between adjacent gates. His original departure gate is assigned at random. After waiting, he is informed that his departure gate has been changed to a different gate, again at random. Calculate the probability that Dave walks $200$ feet or less to the new gate. Express the result as a fraction $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, and find $m+n$.

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