
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 $f(x)= \sqrt {3}\sin 2x+\cos 2x$  
$(1)$ When $x\in[0, \frac {\pi}{2}]$, find the interval of monotonic increase for the function $f(x)$.  
$(2)$ If $x\in[- \frac {\pi}{6}, \frac {\pi}{3}]$, find the range of the function $g(x)= \frac {1}{2}f^{2}(x)-f(x+ \frac {\pi}{4})-1$.

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