
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
Let the functions be $f(x)=x\cdot e^{x}$, $g(x)=x^{2}+2x$, $h(x)=2\sin \left( \frac {\pi}{6}x+ \frac {2\pi}{3}\right)$. If for any $x\in\mathbb{R}$, the inequality $h(x)-f(x)\leqslant k[g(x)+2]$ holds, determine the range of the real number $k$.

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