
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 $f(x)=\cos (2x-\frac{\pi}{3})+\sin^2x-\cos^2x+\sqrt{2}$.

(Ⅰ) Find the smallest positive period of the function $f(x)$ and its intervals of monotonic increase;

(Ⅱ) If there exists $t\in\left[\frac{\pi}{12},\frac{\pi}{3}\right]$ satisfying $[f(t)]^2-2\sqrt{2}f(t)-m > 0$, find the range of the real number $m$.

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