
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 vectors $\overrightarrow{m} = (2\cos x + 2\sqrt{3}\sin x, 1)$ and $\overrightarrow{n} = (\cos x, -y)$, where $x, y \in \mathbb{R}$.
1. If $\overrightarrow{m} \parallel \overrightarrow{n}$ and $y=1$, find the value of $\tan(x + \frac{\pi}{6})$;
2. If $\overrightarrow{m} \perp \overrightarrow{n}$, let $y=f(x)$, find the intervals of increase for the function $f(x)$.

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