<|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
 For each positive integer \( n \), define the point \( P_{n} \) to have coordinates \(\left((n-1)^{2}, n(n-1)\right)\) and the point \( Q_{n} \) to have coordinates \(\left((n-1)^{2}, 0\right)\). For how many integers \( n \) with \( 2 \leq n \leq 99 \) is the area of trapezoid \( Q_{n} P_{n} P_{n+1} Q_{n+1} \) a perfect square? 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
6
For each positive integer \( n \), define the point \( P_{n} \) to have coordinates \(\left((n-1)^{2}, n(n-1)\right)\) and the point \( Q_{n} \) to have coordinates \(\left((n-1)^{2}, 0\right)\). For how many integers \( n \) with \( 2 \leq n \leq 99 \) is the area of trapezoid \( Q_{n} P_{n} P_{n+1} Q_{n+1} \) a perfect square?
test