
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 $f(x)=ex-a\ln x$ where $a\in \mathbb{R}$ attains a local minimum at $x=\frac{1}{e}$, find:<br/>
$(1)$ the value of the real number $a$;<br/>
$(2)$ the range of values for $m$ if there exists $x_{0}$ in the interval $[\frac{1}{e}, e]$ such that the inequality $f(x) \lt x+m$ holds.

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