
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 $f(x)= \sqrt {2}\cos (2x- \frac {\pi}{4})$, where $x\in\mathbb{R}$.  
$(1)$ Find the smallest positive period and the intervals of monotonic increase for the function $f(x)$.  
$(2)$ Find the minimum and maximum values of the function $f(x)$ in the interval $\left[- \frac {\pi}{8}, \frac {\pi}{2}\right]$.

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