
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
The graph of the function $f(x)=2\cos 2x$ is translated to the right by $\dfrac{\pi }{6}$ units to obtain the graph of the function $g(x)$. If the function $g(x)$ is monotonically increasing in the intervals $\left[0,\dfrac{a}{3}\right]$ and $\left[2a,\dfrac{7\pi }{6}\right]$, determine the range of the real number $a$.

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