
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 $f(x) = x \ln x$,
(1) If the inequality $f(x) > kx - \frac{1}{2}$ holds for any positive real number $x$, find the range of real numbers $k$.
(2) Is there an integer $m$ such that for any positive real number $x$, the inequality $f(m + x) < f(m) e^x$ always holds? If it exists, find the smallest integer $m$; if not, explain why.

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