
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 $f(x) = \sqrt{2}\cos\left(2x - \frac{\pi}{4}\right)$, where $x \in \left[-\frac{\pi}{8}, \frac{\pi}{2}\right]$,
(1) Determine the monotonic intervals of the function $f(x)$.
(2) Find the minimum and maximum values of the function $f(x)$ on the interval $\left[-\frac{\pi}{8}, \frac{\pi}{2}\right]$, and obtain the values of $x$ when these extrema are reached.

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