
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 the function $f(x) = \frac{a(x-1)}{x^2}$, where $a \neq 0$,  
(I) Determine the intervals of monotonicity for $f(x)$.  
(II) If the line $x-y-1=0$ is the tangent to the curve $y=f(x)$, find the value of the real number $a$.  
(III) Let $g(x) = x \ln x - x^2 f(x)$, find the minimum value of $g(x)$ on the interval $[1, e]$ (where $e$ is the base of the natural logarithm).

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