
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 $f(x)=\frac{\ln{x}+k}{e^{x}}$ (where $k\in\mathbb{R}$, $e=2.71828...$ is the base of the natural logarithm), and $f′(x)$ is the derivative of $f(x)$.
1. If $f′(1)=0$, find the maximum value of the function $g(x)=f(x)e^{x}-x$.
2. If the equation $f′(x)=0$ has a solution when $x\in(0,1]$, find the range of values for the real number $k$.

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