
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 $f(x)=\sin \left( \frac {5\pi}{6}-2x\right)-2\sin \left(x- \frac {\pi}{4}\right)\cos \left(x+ \frac {3\pi}{4}\right).$  
$(1)$ Find the minimum positive period and the intervals of monotonic increase for the function $f(x)$;  
$(2)$ If $x_{0}\in\left[ \frac {\pi}{3}, \frac {7\pi}{12}\right]$ and $f(x_{0})= \frac {1}{3}$, find the value of $\cos 2x_{0}.$

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