
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
We agree that if the major and minor axes of an ellipse are respectively the real and imaginary axes of another hyperbola, then they are called "sister" conic sections. Given the ellipse ${C_1}:\frac{{{x^2}}}{4}+\frac{{{y^2}}}{{{b^2}}}=1(0＜b＜2)$, the hyperbola $C_{2}$ is the "sister" conic section of the ellipse $C_{1}$, $e_{1}$ and $e_{2}$ are the eccentricities of $C_{1}$ and $C_{2}$, and ${e_1}{e_2}=\frac{{\sqrt{15}}}{4}$, points $M$ and $N$ are the left and right vertices of the ellipse $C_{1}$.  
$(1)$ Find the equation of the hyperbola $C_{2}$;  
$(2)$ Let the moving line $l$ passing through the point $G\left(4,0\right)$ intersect the right branch of the hyperbola $C_{2}$ at points $A$ and $B$, and the slopes of the lines $AM$ and $BN$ are $k_{AM}$ and $k_{BN}$.  
$(i)$ Investigate whether the ratio $\frac{{{k_{AM}}}}{{{k_{BN}}}}$ is a constant. If it is a constant, find this constant; if it is not a constant, explain the reason;  
$(ii)$ Find the range of values for $w=k_{AM}^2+\frac{2}{3}{k_{BN}}$.

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