
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
In his book "On the Sphere and Cylinder," Archimedes derived an equivalent trigonometric identity $\sin \frac{\pi}{2 n}+\sin \frac{2 \pi}{2 n}+\ldots+\sin \frac{(2 n-1) \pi}{2 n}=\frac{1}{\tan \frac{\pi}{4 n}}$ while proving the formula for the volume of a sphere. If both sides are multiplied by $\frac{\pi}{2 n}$ and $n$ approaches $+\infty$, the left side becomes $\lim _{x \rightarrow \infty} \sum_{i=1}^{2 n} \frac{\pi}{2 n} \sin \frac{i \pi}{2 n}=\int_{0}^{\pi} \sin x d x$. Therefore, Archimedes essentially obtained an equivalent result of the definite integral $\int_{0}^{\pi} \sin x d x$. Hence, $\int_{0}^{\pi} \sin x d x=$ \_\_\_\_\_\_.

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