
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 $\overrightarrow{a} = (\cos \frac{3x}{2}, \sin \frac{3x}{2})$ and $\overrightarrow{b} = (\cos \frac{x}{2}, -\sin \frac{x}{2})$, and the function $f(x) = \overrightarrow{a} \cdot \overrightarrow{b} - m|\overrightarrow{a} + \overrightarrow{b}| + 1$, where $x \in [-\frac{\pi}{3}, \frac{\pi}{4}]$ and $m \in \mathbb{R}$.
1. When $m=0$, find the value of $f(\frac{\pi}{6})$;
2. If the minimum value of $f(x)$ is $-1$, find the value of the real number $m$;
3. Determine whether there exists a real number $m$ such that the function $g(x) = f(x) + \frac{24}{49}m^2$, where $x \in [-\frac{\pi}{3}, \frac{\pi}{4}]$, has four distinct zeros. If it exists, find the range of $m$; if not, explain why.

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