
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 a hyperbola $C: \frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1(a>0,b>0)$, where ${F}_{1}$ and ${F}_{2}$ are the left and right foci of $C$. If the eccentricity of $C$ is $e=2$, and the point $(4,6)$ lies on $C$. <br/>$(1)$ Find the equation of $C$. <br/>$(2)$ If a line $l$ passing through $F_{2}$ intersects the left and right branches of $C$ at points $A$ and $B$ (different from the vertices of the hyperbola), is $|\frac{1}{|A{F}_{2}|}-\frac{1}{|B{F}_{2}|}|$ a constant value? If yes, find the constant value; if not, explain the reason.

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