
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
Translate the given function $f(x)=3\sin (2x+\frac{\pi }{3})$ left by $\frac{\pi }{6}$ units, then up by 1 unit to obtain the graph of $g(x)$, and derive an expression for $g(x)$. If $g(x_{1})g(x_{2})=16$, where $x_{1},x_{2}\in [-\frac{3\pi }{2},\frac{3\pi }{2}]$, find the maximum value of $2x_{1}-x_{2}$.

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