
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
If $x \in \left[-\frac{5\pi}{12}, -\frac{\pi}{3}\right]$, determine the maximum value of $y = \tan \left(x + \frac{2\pi}{3}\right) - \tan \left(x + \frac{\pi}{6}\right) + \cos \left(x + \frac{\pi}{6}\right)$.

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