<|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
 The hyperbola \\(C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)\\) has its left and right foci as \\(F_1\\) and \\(F_2\\) respectively, with a focal distance of \\(2c\\). A circle with the right vertex \\(A\\) as its center and a radius of \\(\frac{a+c}{2}\\) passes through \\(F_1\\) and is tangent to line \\(l\\) at point \\(N\\). Let the intersection points of \\(l\\) with \\(C\\) be \\(P\\) and \\(Q\\). If \\(\overrightarrow{PQ} = 2 \overrightarrow{PN}\\), then the eccentricity of the hyperbola \\(C\\) is \_\_\_\_\_\_. 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
The hyperbola \\(C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)\\) has its left and right foci as \\(F_1\\) and \\(F_2\\) respectively, with a focal distance of \\(2c\\). A circle with the right vertex \\(A\\) as its center and a radius of \\(\frac{a+c}{2}\\) passes through \\(F_1\\) and is tangent to line \\(l\\) at point \\(N\\). Let the intersection points of \\(l\\) with \\(C\\) be \\(P\\) and \\(Q\\). If \\(\overrightarrow{PQ} = 2 \overrightarrow{PN}\\), then the eccentricity of the hyperbola \\(C\\) is \_\_\_\_\_\_.
test