
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 $$\overrightarrow{m} = (\cos x + \sqrt{3}\sin x, 1)$$ and $$\overrightarrow{n} = (2\cos x, a)$$ (where $x, a \in \mathbb{R}$, $a$ is a constant),  
(1) Find the function relationship $y = \overrightarrow{m} \cdot \overrightarrow{n}$ with respect to $x$, denoted as $y = f(x)$;  
(2) Determine the interval of monotonic increase for $f(x)$;  
(3) If $x \in [0, \frac{\pi}{2}]$ and the maximum value of $f(x)$ is 4, find the value of $a$.

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