
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 vectors \(\vec{a} = (\sin x, 2 \cos x)\) and \(\vec{b} = (\sqrt{3} \cos x - \sin x, \cos x)\), and \(f(x) = 2 \vec{a} \cdot \vec{b} - 1\):  
(1) If the equation \(f^{2}(x)-(m+3) f(x)+3 m=0\) has four distinct real roots in \([0, \pi]\), find the value of \(m\).   
(2) Do there exist a real number \(a\) and a positive integer \(n\) such that the function \(h(x) = f(x) - a\) has exactly 2023 zeros in the interval \([0, n \pi]\)? If they exist, find all possible values of \(a\) and \(n\); if not, provide an explanation.

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