
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 a function $f(x) = (\sin x + \cos x)^{2} + 2\cos^{2}x - 2$,
(1) Find the smallest positive period $T$ of the function $f(x)$;
(2) Find the maximum value of $f(x)$, and determine the set of $x$ for which this maximum is obtained;
(3) When $x \in \left[ \frac{\pi}{4}, \frac{3\pi}{4} \right]$, find the range of the function $f(x)$.

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