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

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