<|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 that $F_{1}$ and $F_{2}$ are the left and right foci of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ ($a > 0, b > 0$) respectively, and $P$ is a point on the hyperbola. If $\angle F_{1}PF_{2} = 90^{\circ}$ and the lengths of the sides of $\triangle F_{1}PF_{2}$ form an arithmetic sequence, then the eccentricity of the hyperbola 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
5
Given that $F_{1}$ and $F_{2}$ are the left and right foci of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ ($a > 0, b > 0$) respectively, and $P$ is a point on the hyperbola. If $\angle F_{1}PF_{2} = 90^{\circ}$ and the lengths of the sides of $\triangle F_{1}PF_{2}$ form an arithmetic sequence, then the eccentricity of the hyperbola is \_\_\_\_\_\_.
test