
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 $y=f(x)$ ($x \in \mathbb{R}$), for the function $y=g(x)$ ($x \in I$), the "symmetric function" of $g(x)$ with respect to $f(x)$ is defined as $y=h(x)$ ($x \in D, y=h(x)$) satisfying: for any $x \in I$, the two points $(x,h(x))$ and $(x,g(x))$ are symmetric with respect to the point $(x,f(x))$. If $h(x)=\frac{1}{3}\sin 2x$ is the "symmetric function" of $g(x)$ with respect to $f(x)=\frac{1}{2}x+\frac{1}{2}a\sin x$, and $g(x)$ is a monotonically increasing function on $(-\infty,+\infty)$, then the range of the real number $a$ is ______.

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