
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 $f(x)=2\cos x(\sin x+\cos x)$, where $x \in \mathbb{R}$.

(1) Find the smallest positive period of the function $f(x)$.

(2) Find the intervals on which the function $f(x)$ is monotonically increasing.

(3) Find the minimum and maximum values of the function $f(x)$ on the interval $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$.

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