
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 $a=(a\_1,a\_2)$ and $b=(b\_1,b\_2)$, define a new operation: "$a \cdot b\_1 = (a\_1 b\_1, a\_2 b\_2)$". Point $P$ moves on the curve $y=\sin x$ and point $Q$ moves on the curve $y=f(x)$. The position vector of $Q$ is given by $\overrightarrow{OQ} = m \cdot \overrightarrow{OP} + n$, where $O$ is the origin, $m = \left(\begin{matrix} \frac{1}{2}, 3 \end{matrix}\right)$, and $n = \left(\begin{matrix} \frac{\pi}{6}, 0 \end{matrix}\right)$. Find the maximum value of $y=f(x)$.

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