
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 vectors $\overrightarrow{a}=(\sin x,-\cos x)$, $\overrightarrow{b}=( \sqrt {3}\cos x,\cos x)$, let the function $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}- \frac {1}{2}$;  
$(1)$ Write down the interval of monotonic increase for the function $f(x)$;  
$(2)$ If $x\in\left[ \frac {\pi}{4}, \frac {\pi}{2}\right]$, find the maximum and minimum values of the function $f(x)$ and the corresponding values of $x$;  
$(3)$ If the inequality $|f(x)-m| < 1$ holds for all $x\in\left[ \frac {\pi}{4}, \frac {\pi}{2}\right]$, find the range of the real number $m$．

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