
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
A particle moves such that its speed for each subsequent mile varies inversely as the square of the integral number of miles already traveled. If the speed for the first mile was $1$ mile per hour and the second mile was traversed in $2$ hours, determine the time in hours needed to traverse the $n$-th mile.

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