
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
If each point on the graph of the function $y=f(x)$ has its vertical coordinate remain unchanged, its horizontal coordinate stretched to twice its original length, and then the entire graph is shifted left by $\frac{\pi}{3}$ units along the $x$-axis, and finally, the resulting graph is shifted down by $1$ unit along the $y$-axis, resulting in the graph of the function $y=\frac{1}{2}\sin x$, then the analytical expression of the function $f(x)$ is \_\_\_\_\_\_.

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