
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 $f(x)=2\sin\left(2x+\frac{\pi}{6}\right)$, find the maximum value of $2x_1-x_2$ if $g(x_1)g(x_2)=9$, $x_1$, $x_2\in[-2\pi,2\pi]$, where $g(x)$ is the function obtained by shifting $f(x)$ left by $\frac{\pi}{12}$ units and then up by 1 unit.

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