
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
Given the following 8 types of graph transformations:

1. The vertical coordinates of all points on the graph remain unchanged, and the horizontal coordinates are shortened to half of the original;
2. The vertical coordinates of all points on the graph remain unchanged, and the horizontal coordinates are extended to twice the original;
3. The horizontal coordinates of all points on the graph remain unchanged, and the vertical coordinates are shortened to half of the original;
4. The horizontal coordinates of all points on the graph remain unchanged, and the vertical coordinates are extended to twice the original;
5. The graph is shifted to the right by $\frac{\pi}{3}$ units;
6. The graph is shifted to the left by $\frac{\pi}{3}$ units;
7. The graph is shifted to the right by $\frac{2\pi}{3}$ units;
8. The graph is shifted to the left by $\frac{2\pi}{3}$ units.

Select some of the transformation methods above and arrange them in a certain order to transform the graph of the function $y=\sin x$ into the graph of the function $y=\frac{1}{2}\sin (\frac{x}{2}+\frac{\pi}{3})$. Write out the analytical expression of the function obtained after each transformation (only one method is required).

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