
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
Suppose the function $f(x)$ has a derivative $f′(x)$ that also has a derivative, denoted as $f^{n}(x)$. If $f(x)$ satisfies $f^{n}(x) < 0$ for any $x∈(a,b)$, then $f(x)$ has the following property:
$$f\left( \frac {x_{1}+x_{2}+…+x_{n}}{n}\right)\geqslant \frac {f(x_{1})+f(x_{2})+…+f(x_{n})}{n}$$
where $n∈N^{*}$, $x_{1}$, $x_{2}$, $…$, $x_{n}∈(a,b)$. If $f(x)=\sin x$, then $f^{n}(x)=$ \_\_\_\_\_\_ ; according to the above property, when $x_{1}+x_{2}+x_{3}=π$ and $x_{1}$, $x_{2}$, $x_{3}∈(0,π)$, infer the maximum value of $\sin x_{1}+\sin x_{2}+\sin x_{3}$ is \_\_\_\_\_\_.

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