
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 $f(x)=axe^{x}$, where $a\neq 0$, and $e$ is the base of the natural logarithm.
(I) Find the monotonic intervals of the function $f(x)$;
(II) When $a=1$, find the extreme values of the function $f(x)$;
(III) If the line $y=e(x-\frac{1}{2})$ is a tangent line of the curve $y=f(x)$, find the value of the real number $a$.

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