
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 so that its speed for the third and subsequent miles varies inversely as the square of the integral number of miles already traveled and directly as the time squared. For the third mile, the time taken is 3 seconds. Determine the time, in seconds, needed to traverse the \( n \)th mile for \( n \geq 3 \).
A) \( \sqrt[3]{\frac{27(n-1)^2}{4}} \) seconds
B) \( \frac{27(n-1)}{4} \) seconds
C) \( \sqrt{\frac{27(n-1)^2}{4}} \) seconds
D) \( \frac{9(n-1)^2}{4} \) seconds

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