
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 a function $f(x)=\frac {a(x-1)}{x^{2}}$, where $a>0$,
(1) Determine the intervals of monotonicity for the function $f(x)$;
(2) 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 natural logarithms).

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