
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 functions $f(x)= \frac {1}{2}x^{2}+ax+2\ln x$ and $g(x)= \frac {1}{2}x^{2}+kx+(2-x)\ln x-k$, where $k\in\mathbb{Z}$.
1. When $a=-3$, find the intervals of monotonicity for $f(x)$.
2. When $a=1$, if for any $x > 1$, $g(x) < f(x)$ holds, find the maximum value of $k$.

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