Example 4 Find the smallest positive period of the function $y=\left|2 \sin \left(x+\frac{\pi}{3}\right)+1\right|$.

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Analysis From the functional form given, the period is not very clear, and it may even lead to misunderstandings. If special values are used to test, it seems difficult to determine. However, the graph of this function is relatively easy to draw. For such functions, the graphical method is more accurate and convenient for finding the period.

Solution The graph of the function $y=\left|2 \sin \left(x+\frac{\pi}{3}\right)+1\right|$ can be obtained through the following transformations: First, shift the graph of $y=\sin x$ to the left by $\frac{\pi}{3}$ units, then stretch all points on the curve vertically to twice their original length (keeping the horizontal coordinates unchanged), then shift the resulting graph upward by 1 unit, and finally reflect the part of the graph below the $x$-axis to above the $x$-axis. The graph is shown in Figure 2-1-2. Observing the graph, we can see that the smallest positive period of the function is $2 \pi$.