
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 $f(x)=A\cos(\omega x + \varphi)$ defined on $\mathbb{R}$, with $A>0$, $\omega>0$, and $|\varphi| \leq \frac{\pi}{2}$, satisfies the following conditions: the maximum value is 2, the distance between two adjacent minimal points of the graph is $\pi$, and the graph of $f(x)$ is symmetric about the point $(\frac{\pi}{12}, 0)$.
1. Determine the analytic expression of $f(x)$;
2. Let $\vec{a} = (f(x - \frac{\pi}{6}), 1)$ and $\vec{b} = (\frac{1}{2}, -2\cos x)$, where $x \in \left[-\frac{3\pi}{4}, \frac{\pi}{2}\right]$. Define $g(x) = \vec{a} \cdot \vec{b} + \frac{1}{2}$. Find the range of the function $g(x)$.

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