
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 $f(x)= \frac {1}{2}\cos2x+ \sqrt {3}\sin x\cos x-2\cos^{2}x$,  
(1) find the maximum value of $f(x)$;  
(2) if $f(a)=- \frac {1}{5}$, find the value of $\cos(2a+ \frac {\pi }{3})$;  
(3) if $\frac {\pi }{3}<a<\frac {\pi }{2}$ and $f(a)=- \frac {1}{5}$, find the value of $\cos2a$.

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