<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 Given the hyperbola x^2/a^2 - y^2/b^2 = 1 (a > 0, b > 0) with its right focus at F(2,0), let A and B be two points on the hyperbola symmetric about the origin. The midpoint of AF is M, and the midpoint of BF is N. If the origin O lies on the circle with diameter MN, and the slope of line AB is 3*sqrt(7)/7, calculate the eccentricity of the hyperbola. Express your answer as a single numerical value. Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
2
Given the hyperbola x^2/a^2 - y^2/b^2 = 1 (a > 0, b > 0) with its right focus at F(2,0), let A and B be two points on the hyperbola symmetric about the origin. The midpoint of AF is M, and the midpoint of BF is N. If the origin O lies on the circle with diameter MN, and the slope of line AB is 3*sqrt(7)/7, calculate the eccentricity of the hyperbola. Express your answer as a single numerical value.
test