
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 the length of the interval $[x\_1, x\_2]$ as $x\_2 - x\_1 (x\_2 > x\_1)$, and the domain and range of the function $f(x) = \frac{(a^2 + a)x - 1}{a^2x} (a \in \mathbb{R}, a \neq 0)$ are both $[m, n]$. Find the value of $a$ when the interval $[m, n]$ has the maximum length.

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