
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 $f(x) = \frac{1}{3}x^{3} - \frac{1}{2}(a+2)x^{2} + x \ (a \in \mathbb{R})$
(1) When $a=0$, let $k$ be the slope of the tangent line to the curve of $f(x)$ at point $P$. Find the minimum value of $k$.
(2) Let $g(x) = e - \frac{e^{x}}{x}$ (where $e$ is the base of the natural logarithm). If $f'(x) \geq g(x)$ holds true for all $x > 0$, find the range of values for the real number $a$.

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