
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 functions $f(x)= \frac {\ln x}{x}$ and $g(x)=x(\ln x- \frac {ax}{2}-1)$.  
(1) Find the maximum value of $y=f(x)$.  
(2) When $a\in[0, \frac{1}{e}]$, find the minimum value of the function $y=g(x)$ for $x\in(0,e]$. Let the minimum value of $g(x)$ be denoted as $h(a)$, determine the range of the function $h(a)$.

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