
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 the function $f(x)=\cos x(\sin x+\cos x)- \frac {1}{2}$.  
$(1)$ If $0 < \alpha < \frac {\pi}{2}$, and $\sin \alpha= \frac { \sqrt {2}}{2}$, find the value of $f(\alpha)$;  
$(2)$ Find the smallest positive period and the intervals of monotonic increase for the function $f(x)$.

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