
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 $\overrightarrow{a}=(\sin x,\cos x)$ and $\overrightarrow{b}=(\cos (x+ \frac {\pi}{6})+\sin x,\cos x)$, and the function $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$.
- (I) Find the interval of monotonic increase for $f(x)$;
- (II) If $\alpha\in(0, \frac {\pi}{2})$ and $\cos (\alpha+ \frac {\pi}{12})= \frac {1}{3}$, find $f(\alpha)$.

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