
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 $f(x)=\ln (x+1)-x$,
(1) If $k\in\mathbb{Z}$, and $f(x-1)+x > k(1- \frac {3}{x})$ holds for any $x > 1$, find the maximum value of $k$.
(2) For any constant $a$ in $(0,1)$, does there exist a positive number $x_{0}$, such that $e^{f(x_{0})} < 1- \frac {a}{2}x_{0}^{2}$ holds.

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