
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 $f(x)=\frac{ex}{{e}^{x}}$ (where $e$ is the base of the natural logarithm).

(1) Find the intervals of monotonicity for $f(x)$;

(2) Determine whether there exists a positive real number $x$ such that $f(1-x)=f(1+x)$; if it exists, find $x$, otherwise explain why.

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