
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 $f(x) = \ln(e^x + k)$, where $k$ is a constant and $e$ is the base of the natural logarithm, it is known that the function is an odd function on the set of real numbers $\mathbb{R}$.
(I) Find the value of $k$;
(II) Discuss the number of roots of the equation $\frac{\ln x}{f(x)} = x^2 - 2ex + m$ with respect to $x$.

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