
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 $p=(1, \sqrt{3})$ and $q=(\cos x,\sin x)$,

    (1) If $p \parallel q$, find the value of $\sin 2x-\cos^2x$;

    (2) Let the function $f(x)=p\cdot q$, scale down all the x-coordinates of the points on the graph of the function to half (the y-coordinates remain unchanged), and then shift the resulting graph to the left by $\frac{\pi}{3}$ units to get the graph of the function $g(x)$. Find the intervals of increase for $g(x)$.

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