
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 vectors $\overrightarrow{a} = (2\sin x, 1)$ and $\overrightarrow{b} = (\cos x, 1-\cos 2x)$, and the function $f(x) = \overrightarrow{a} \cdot \overrightarrow{b}$ ($x \in \mathbb{R}$).

(1) Find the smallest positive period, the maximum value, and the minimum value of the function $f(x)$;  
(2) Find the intervals where $f(x)$ is monotonically increasing.

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