
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 $\overrightarrow {a}$ = (${\sqrt {3}\cos{\omega x}}$, $\cos{\omega x}$) and $\overrightarrow {b}$ = ($\sin{\omega x}$, $-\cos{\omega x}$), the minimum distance between the two symmetry centers of the function $f(x) = \overrightarrow {a} \cdot \overrightarrow {b}$ is $\frac{\pi}{2}$.

1. Find $f(\frac{\pi}{3})$.
2. If the function $G(x) = m + 1 - \sqrt{2}f(\frac{x}{2})$ has exactly two zero points in the interval $[0, \pi]$, find the range of real numbers $m$.

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