
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
Given a point $P_{0}\left(x_{0}, y_{0}\right)$ and a line with the equation $ax + by = 0 \ (a \neq 0)$. Let $Q_{1}$ be the point symmetric to $P_{0}$ with respect to the $x$-axis. Draw a line through $Q_{1}$ parallel to $l$, intersecting the $x$-axis at $R_{1}$. Let $P_{1}$ be the point symmetric to $Q_{1}$ with respect to $R_{1}$. Similarly, construct $Q_{2}$, $P_{2}$, and then $Q_{3}$, $P_{3}$, and so on. If we denote the coordinates of point $P_{n}$ by $\left(x_{n}, y_{n}\right)$, under what conditions do the sequences $\left\{x_{n}\right\}$ and $\left\{y_{n}\right\}$ converge? What are their limit values if they converge?

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