
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
The property theorem of convex functions states: If the function $f(x)$ is convex on interval $D$, then for any $x_1, x_2, \cdots, x_n$ within interval $D$, it holds that $\frac{f(x_1)+f(x_2)+\cdots+f(x_n)}{n} \leqslant f(\frac{x_1+x_2+\cdots+x_n}{n})$. Given that the function $y=\sin x$ is convex on the interval $(0,\pi)$, the maximum value of $\sin A+\sin B+\sin C$ in $\triangle ABC$ is $\boxed{\frac{3\sqrt{3}}{2}}$.

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