
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 $f(x)=A\sin (\omega x+\varphi)+B$ ($A > 0$, $\omega > 0$, $|\varphi| < \frac {\pi}{2}$) has a maximum value of $2 \sqrt {2}$, a minimum value of $- \sqrt {2}$, a period of $\pi$, and the graph passes through the point $\left(0,- \frac { \sqrt {2}}{4}\right)$.  
$(1)$ Find the expression for the function $f(x)$ and the intervals of monotonic increase for $f(x)$.  
$(2)$ If the equation $f(x)=a$ has two roots $\alpha$ and $\beta$ in the interval $\left[0, \frac {7\pi}{12}\right]$, find the value of $\alpha+\beta$ and the range of $a$.

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