
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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