
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 that $f(e^{x})=ax^{2}-x$, where $a\in R$.
(1) Find the analytic expression of $f(x)$;
(2) Find the range of $f(x)$ when $x\in(0,1]$;
(3) If $a > 0$, and $h(x)=[f(x)+1-a]\cdot\log_{x}e$ satisfies $|h(x_{1})-h(x_{2})|\leqslant a+ \frac{1}{3}$ for any $x_{1}$, $x_{2}\in[e^{-3},e^{-1}]$, find the range of real number $a$.

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