
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
Let ellipse \(C_1: \frac{x^2}{16}+\frac{y^2}{12}=1\) and parabola \(C_2: y^2=8 x\) intersect at point \(P(x_0, y_0)\). Define the function:
\[ 
f(x)=\left\{
\begin{array}{ll}
2 \sqrt{2 x}, & 0<x<x_{0}, \\
\frac{\sqrt{3}}{2} \cdot \sqrt{16-x^{2}}, & x>x_{0},
\end{array}
\right.
\]

If the line \(y=a\) intersects \(f(x)\) at points \(A\) and \(B\), and considering the fixed point \(N(2,0)\), determine the range of the perimeter of \(\triangle ABN\).

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