<|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 $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (where $a > 0$, $b > 0$), the left and right vertices are A($-a, 0$) and B($a, 0$) respectively. The equations of the asymptotes are $l_1: y = \frac{b}{a}x$ and $l_2: y = -\frac{b}{a}x$. Point P is in the first quadrant and lies on $l_1$. If $PA$ is perpendicular to $l_2$ and $PB$ is parallel to $l_2$, 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 $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (where $a > 0$, $b > 0$), the left and right vertices are A($-a, 0$) and B($a, 0$) respectively. The equations of the asymptotes are $l_1: y = \frac{b}{a}x$ and $l_2: y = -\frac{b}{a}x$. Point P is in the first quadrant and lies on $l_1$. If $PA$ is perpendicular to $l_2$ and $PB$ is parallel to $l_2$, calculate the eccentricity of the hyperbola. Express your answer as a single numerical value.
test