
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 vectors $\overrightarrow{a}=(a\_1,a\_2)$ and $\overrightarrow{b}=(b\_1,b\_2)$, define a vector product: $\overrightarrow{a}⊗\overrightarrow{b}=(a\_1b\_1,a\_2b\_2)$. It is known that $\overrightarrow{m}=(1,\frac{1}{2})$ and $\overrightarrow{n}=(0,1)$, and point $P(x,y)$ moves on the graph of the function $y=\sin\frac{x}{2}$. Point $Q$ moves on the graph of the function $y=f(x)$, and points $P$ and $Q$ satisfy: $\overrightarrow{OQ}=\overrightarrow{m}⊗\overrightarrow{OP}+\overrightarrow{n}$ (where $O$ is the coordinate origin). Determine the maximum value $A$ and the smallest positive period $T$ of the function $y=f(x)$.

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