
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 $f(x)=A\sin (\omega x+\varphi)+B$ ($A > 0$, $\omega > 0$, $|\varphi| < \frac {\pi}{2}$), its smallest positive period is $2\pi$, the minimum value is $-2$, and when $x= \frac {5\pi}{6}$, the function reaches its maximum value of $4$.  
$(I)$ Find the expression for the function $f(x)$;  
$(II)$ Find the interval of monotonic increase for the function $f(x)$;  
$(III)$ If when $x\in\left[ \frac {\pi}{6}, \frac {7\pi}{6}\right]$ the equation $f(x)=m+1$ has a solution, find the range of values for the real number $m$.

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