<|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 $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ $(a > 0, b > 0)$ with an eccentricity of $\frac{5}{4}$, its foci are $F_1$ and $F_2$. $P$ is a point on the hyperbola such that $\overrightarrow{PF_1} \cdot \overrightarrow{PF_2} = 0$. If the area of $\triangle PF_1F_2$ is equal to $9$, calculate the value of $a + b$. 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
7
Given the hyperbola $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ $(a > 0, b > 0)$ with an eccentricity of $\frac{5}{4}$, its foci are $F_1$ and $F_2$. $P$ is a point on the hyperbola such that $\overrightarrow{PF_1} \cdot \overrightarrow{PF_2} = 0$. If the area of $\triangle PF_1F_2$ is equal to $9$, calculate the value of $a + b$.
test