
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
On a Cartesian coordinate plane, points \( P(x, y) \) and \( P^{\prime}\left(x^{\prime}, y^{\prime}\right) \) correspond to each other and satisfy the following system of equations:
\[
\left\{
\begin{array}{l}
x^{\prime}=a x-b y+p, \\
y^{\prime}=b x+a y+q,
\end{array}
\right.
\]
where \( a \), \( b \), \( p \), and \( q \) are constants, with \( b \neq 0 \). Suppose we have two pairs of corresponding points \( P_{1}\left(x_{1}, y_{1}\right) \) and \( P_{1}^{\prime}\left(x_{1}^{\prime}, y_{1}^{\prime}\right) \) as well as \( P_{2}\left(x_{2}, y_{2}\right) \) and \( P_{2}^{\prime}\left(x_{2}^{\prime}, y_{2}^{\prime}\right) \). Consider the following questions:

1. What is the relationship between \( \left|P_{1} P_{2}\right| \) and \( \left|P_{1}^{\prime} P_{2}^{\prime}\right| \)?
2. If a point \( Q \) corresponds to itself, what are the coordinates of point \( Q \)?
3. What is the similarity relationship between triangles \( \triangle Q P_{1} P_{2} \) and \( \triangle Q P_{1}^{\prime} P_{2}^{\prime} \)? Also, determine the ratio of their areas.

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