
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 $f(x)=\sin x\cos \left( x+\dfrac{\pi }{6} \right)$, $x\in R$.

(1) The graph of $f(x)$ is translated to the right by $\dfrac{\pi }{6}$ units to obtain the graph of $g(x)$. Find the interval(s) where $g(x)$ is monotonically increasing.

(2) If $f(\alpha )=-\dfrac{5}{12}$, and $0 < \alpha < \dfrac{\pi }{2}$, find the value of $\sin 2\alpha$.

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