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

(1) Find the intervals where $f(x)$ is monotonically increasing/decreasing and its maximum value.

(2) If for any $x \in (0, +\infty)$, $2|\ln x - \ln 2| \geq f(x) + c - \frac{1}{e^2}$ always holds, find the range of values for the real number $c$.

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