
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 $f(x)=\sqrt{2}\cos(2x-\frac{π}{4})$, $x\in R$.  
$(1)$ Find the smallest positive period and the interval of monotonic increase of the function $f(x)$.  
$(2)$ Find the minimum and maximum values of the function $f(x)$ on the interval $[-\frac{π}{8},\frac{π}{2}]$, and determine the values of $x$ at which these extrema occur.

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