
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 the hyperbola $C$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$), its left vertex is $A$, and point $B\left( 0,\frac{\sqrt{15}}{3}b \right)$. If the perpendicular bisector of segment $AB$ passes through the right focus $F$, calculate the eccentricity of the hyperbola $C$.

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