
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
Convex functions are an important class of functions, characterized by the following property: for any $x_{i} \in (a,b) (i=1,2,\ldots,n)$, it must hold that $f\left( \frac{x_{1}+x_{2}+\ldots+x_{n}}{n}\right) \geqslant \frac{f(x_{1})+f(x_{2})+\ldots+f(x_{n})}{n}$. Given that $y=\sin x$ is a convex function on $(0,\pi)$, using the property of convex functions, when the circumradius of $\Delta ABC$ is $R$, the maximum value of its perimeter is ______.

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