
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 $\overrightarrow{m}=(2\sin x,2\cos x)$, $\overrightarrow{n}=(\cos \frac {\pi}{3},-\sin \frac {\pi}{3})$, $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}+1$.

(Ⅰ) Find the value of $f\left( \frac {\pi}{2}\right)$ and the maximum value of $f(x)$;

(Ⅱ) If the function $g(x)=f\left( \frac {\pi}{2}x\right)$, find the sum $g(1)+g(2)+g(3)+…+g(2014)+g(2015)$;

(Ⅲ) If the function $h(x)= \frac {\sin x\cdot f^{2}\left(x+ \frac {\pi}{3}\right)-8}{1+\cos ^{2}x}$ has a maximum value of $M$ and a minimum value of $m$ in the interval $\left[- \frac {5\pi}{4}, \frac {5\pi}{4}\right]$, find the value of $M+m$.

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