
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 $f(x)=3\sin(2x-\frac{π}{3})-2\cos^{2}(x-\frac{π}{6})+1$, the graph of the function $f(x)$ is shifted to the left by $\frac{π}{6}$ units, resulting in the graph of the function $g(x)$. Find $\sin (2x_{1}+2x_{2})$, where $x_{1}$ and $x_{2}$ are the two roots of the equation $g(x)=a$ in the interval $[0,\frac{π}{2}]$.

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