
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 $\overrightarrow{a} = (1 + \cos \omega x, 1)$ and $\overrightarrow{b} = (1, a + \sqrt{3} \sin \omega x)$ (where $\omega$ is a constant and $\omega > 0$), the function $f(x) = \overrightarrow{a} \cdot \overrightarrow{b}$ has a maximum value of $2$ on $\mathbb{R}$.  
(1) Determine the value of the real number $a$;  
(2) By translating the graph of the function $y = f(x)$ to the right by $\frac{\pi}{6\omega}$ units, the graph of the function $y = g(x)$ is obtained. If $y = g(x)$ is increasing on the interval $[0, \frac{\pi}{4}]$, find the maximum value of $\omega$.

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