
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 the function $f(x) = e^{x-m} - x$, where $m \in \mathbb{R}$.

(I) Find the extremum of the function $f(x)$.

(II) Given the theorem: If the function $y=f(x)$ is continuous on the interval $[a, b]$ and $f(a) \cdot f(b) < 0$, then the function $y=f(x)$ has a root in the interval $(a, b)$, i.e., there exists $x \in (a, b)$ such that $f(x) = 0$. Determine whether the function $f(x)$ has a root in the interval $(m, 2m)$ when $m > 1$.

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