
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 $\overrightarrow{a} = (\sin x, \cos x)$, $\overrightarrow{b} = (\sqrt{3}\cos x, \cos x)$, and $\overrightarrow{b} \neq 0$, define the function $f(x) = 2\overrightarrow{a} \cdot \overrightarrow{b} - 1$.
(1) Find the interval of monotonic increase for the function $f(x)$;
(2) If $\overrightarrow{a} \parallel \overrightarrow{b}$, find the value of $\tan x$;
(3) If $\overrightarrow{a} \perp \overrightarrow{b}$, find the smallest positive value of $x$.

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