
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
Given that a particle's speed for the second and subsequent miles varies inversely as the integral number of miles already traveled, compounded by an exponentially increasing resistance factor that doubles the effect of each previous mile, with the speed \( s_n \) for the \( n \)th mile given by \( s_n = \frac{k}{(n-1)2^{n-2}} \), where the second mile is traversed in 2 hours, find the time needed to traverse the nth mile.

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