
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 function $y=f(x)$, each point on its graph is transformed such that the vertical coordinate remains unchanged, while the horizontal coordinate is doubled. The resulting graph is then shifted left by $\frac{\pi}{2}$ units along the $x$-axis, yielding a curve identical to the graph of $y=\frac{1}{2}\sin(x)$. Determine the analytical expression for $y=f(x)$.

A) $y=\frac{1}{2}\sin\left(\frac{x}{2}-\frac{\pi}{2}\right)$
B) $y=\frac{1}{2}\sin\left(\frac{\pi}{2}-2x\right)$
C) $y=\frac{1}{2}\sin\left(\frac{x}{2}+\frac{\pi}{2}\right)$
D) $y=\frac{1}{2}\sin\left(2x-\frac{\pi}{2}\right)$

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