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

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