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

(I) If $f(x)$ is an increasing function on the interval $(-1, 1)$, find the range of values for $k$;
(II) Determine the minimum value of $f(x)$ on the interval $[0, 1]$;
(III) If $k=0$, does there exist a real number $a$ such that for any $x_1, x_2 \in (a, +\infty)$ with $x_1 < x_2$, the inequality $x_1(f(x_2) - f(a)) - x_2(f(x_1) - f(a)) > a(f(x_2) - f(x_1))$ always holds? If it exists, find the range of values for $a$; if it does not exist, please explain why.

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