
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 that the line $y=x-2$ intersects the hyperbola $C: \frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1 (a>0, b>0)$ at the intersection points $A$ and $B (not coincident)$. The perpendicular bisector of line segment $AB$ passes through the point $(4,0)$. The eccentricity of the hyperbola $C$ is ______.

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