
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 $f(x) = \frac{1}{2}x^{2}-(a+2)x+2a\ln x$ where $a \in \mathbb{R}$.
1. Discuss the monotonicity of the function $f(x)$.
2. If $a=0$, the inequality $kx+f'(x)\ln x > 0$ (where $f'(x)$ is the derivative of $f(x)$ and $k$ is an integer) holds true for all $x$. Find the minimum value of $k$.

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