
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
Let the coordinates of the moving points \( P \) and \( P^{\prime} \) be \( (x, y) \) and \((x^{\prime}, y^{\prime}) \) respectively, and they satisfy the equations:
\[
\left\{
\begin{array}{l}
x^{\prime}=3 x+2 y+1, \\
y^{\prime}=x+4 y-3 .
\end{array}
\right.
\]

If \( P \) and \( P^{\prime} \) move along the same line, does such a line exist? If it exists, find its equation.

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