
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
Let the function \( f(x) = \frac{x^2 + n}{x^2 + x + 1} \) where \( n \in \mathbf{N}^* \), have a maximum value \( a_n \) and a minimum value \( b_n \). Determine \( a_n - b_n \).

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