
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 $f(x)= \frac { \sqrt {3}}{2}\sin 2x-\cos ^{2}x-m$.  
$(1)$ Find the smallest positive period and the intervals of monotonic increase for the function $f(x)$.  
$(2)$ If $x\in\left[ \frac {5\pi}{24}, \frac {3\pi}{4}\right]$ and the maximum value of the function $f(x)$ is $0$, find the value of the real number $m$.

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