
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 $f(x)=2\sin\left(2x+ \dfrac{\pi}{4}\right) + 2\cos^2\left(x+ \dfrac{\pi}{8}\right) - 1$. If the graph of function $f(x)$ is translated to the right by $\dfrac{\pi}{8}$ units to get the graph of function $g(x)$, determine the value of $\sin\left(x_1+x_2\right)$, where $x_1$, $x_2$ are the two roots of $g(x)-m=0$ within the interval $\left[0, \dfrac{\pi}{2}\right]$.

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