
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 with fifteen gates arranged in a straight line with exactly $90$ feet between adjacent gates. His departure gate is chosen at random. After waiting at that gate, Dave is informed that the departure gate has been changed to another random gate. Calculate the probability that Dave walks $450$ feet or less to the new gate. Express this probability 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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