
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 $f(x)=\frac{{2x}}{{{e^x}}}$, where $(e$ is the base of the natural logarithm, $e\approx 2.718)$:
- $(1)$ Find the extreme values of the function $f(x)$.
- $(2)$ Find the maximum and minimum values of the function $f(x)$ on the interval $[{\frac{1}{2},2}]$.

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