
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 theorem of properties of convex functions states that if a function $f(x)$ is convex on the interval $D$, then for any $x\_1$, $x\_2$, $...$, $x\_n$ within the interval $D$, the following inequality holds: $\frac{f\mathrm{(}x\_{1}\mathrm{)}{+}f\mathrm{(}x\_{2}\mathrm{)}{+}\mathrm{{...}}{+}f\mathrm{(}x\_{n}\mathrm{)}}{n}\leqslant f\left( \frac{x\_{1}{+}x\_{2}{+}\mathrm{{...}}{+}x\_{n}}{n} \right).$ Given that the function $y=\sin x$ is convex on the interval $(0,\pi)$, find the maximum value of $\sin A+\sin B+\sin C$ in $\triangle ABC$.

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