
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 $f(x)=|x^{2}-1|+x^{2}+kx$.
(1) Find the range of values for the real number $k$ such that for any $x$ in the interval $(0,+∞)$, $f(x)\geqslant 0$ holds.
(2) If the function $f(x)$ has two distinct zero points $x_{1}$, $x_{2}$ in the interval $(0,2)$, find:
   $\quad\quad$ (i) the range of values for the real number $k$;
   $\quad\quad$ (ii) the range of values for $\frac {1}{x_{1}}+ \frac {1}{x_{2}}$.

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