
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 $f(x)=\cos (2x- \frac {\pi}{3})+\sin ^{2}x-\cos ^{2}x+ \sqrt {2}$.
$(1)$ Find the smallest positive period of the function $f(x)$ and the intervals of monotonic increase;
$(2)$ If there exists $t\in\left[ \frac {\pi}{12}, \frac {\pi}{3}\right]$ satisfying $\left[f(t)\right]^{2}-2 \sqrt {2}f(t)-m > 0$, find the range of the real number $m$.

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