
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
John arrives at a terminal which has sixteen gates arranged in a straight line with exactly 50 feet between adjacent gates. His departure gate is assigned randomly. After waiting at that gate, John is informed that the departure gate has been changed to another gate, chosen randomly again. What is the probability that John walks 200 feet or less to the new gate? Let this probability be expressed as a fraction $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

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