
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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?

Present the answer in LaTex format: \boxed{Your answer}
user