
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 $\sin (\pi-\alpha)= \frac {4}{5}$, $\alpha\in(0, \frac {\pi}{2}).$  
$(1)$ Find the value of $\sin 2\alpha-\cos ^{2} \frac {\alpha}{2};$  
$(2)$ Find the interval of monotonic increase for the function $f(x)= \frac {5}{6}\cos \alpha\sin 2x- \frac {1}{2}\cos 2x.$

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