
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 $\overrightarrow{a} = (2\cos^2 x, 1)$ and $\overrightarrow{b} = (1, 2\sqrt{3} \sin x \cos x + m)$, where $x \in \mathbb{R}$, $m \in \mathbb{R}$, and $m$ is a constant, and $y = \overrightarrow{a} \cdot \overrightarrow{b}$.

(1) Find the function $y = f(x)$;
(2) If $x \in [0, \frac{\pi}{2}]$, and the maximum value of $f(x)$ is 4, find the value of $m$;
(3) Find the smallest positive period and the monotonically decreasing interval of $f(x)$.

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