
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
Define \( ||x|| \) as the distance from \( x \) to the nearest integer. Find \( \lim_{n \to \infty} \frac{1}{n} \int \limits_{1}^{n} ||\frac{n}{x}|| \, dx \). You may assume that \( \prod_{n=1}^{\infty} \frac{2n}{2n-1} \cdot \frac{2n}{2n+1} = \frac{\pi}{2} \).

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