
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 $f(x) = e^{x-k} - x$, where $x \in \mathbb{R}$,
(1) When $k=0$, if the inequality $f(x) \geq m$ holds true for all real $x$, find the range of real numbers $m$.
(2) Determine whether there are two distinct points in the interval $(k, 2k)$ for which $f(x)$ equals zero when $k > 1$. If such points exist, find the number of zeros.

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