
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 ellipse $C_{1}$: $\frac{x^2}{4}+\frac{y^2}{3}=1$, parabola $C_{2}$: $y^{2}=4x$, one intersection point of the two curves is $A(x_{0}, y_{0})$, point $B(1,0)$. Define $f(x)=\left\{\begin{array}{l}{2\sqrt{x},0<x<x_{0}}\\{\frac{\sqrt{3(4-x^{2})}}{2},x\geq x_{0}}\end{array}\right.$. If the lines $y=a$ and $y=f(x)$ intersect at points $M$ and $N$, then the range of the perimeter of $\triangle BMN$ is ______.

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