
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
**Lina arrives at an airport which has twenty gates arranged in a straight line with exactly 50 feet between adjacent gates. Her departure gate is initially assigned at random. After waiting at that gate, Lina is informed that the departure gate has changed to another randomly selected gate. Determine the probability that Lina walks 200 feet or less to the new gate. Express your answer 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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