
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 $f(x)=\begin{cases} & -x^{2}+x+k,x\leqslant 1 \\\\ & -\frac{1}{2}+\log_{\frac{1}{3}}x,x > 1 \\\\ \end{cases}$ and $g(x)=a\ln (x+2)+\frac{x}{x^{2}+1}(a\in R)$, if for any $x_{1}$, $x_{2}$${x|x\in R,x > -2}$, we have $f(x_{1})\leqslant g(x_{2})$, find the range of values for the real number $k$.

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