
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 \( P_{1} \) be a point on the side \( AB \) of an equilateral triangle \( \triangle ABC \). Draw a perpendicular from \( P_{1} \) to side \( BC \) with the foot of the perpendicular as \( Q_{1} \). Draw a perpendicular from \( Q_{1} \) to side \( CA \) with the foot of the perpendicular as \( R_{1} \). Draw a perpendicular from \( R_{1} \) to side \( AB \) with the foot of the perpendicular as \( P_{2} \), and continue this process to obtain the points \( Q_{2}, R_{2}, P_{3}, Q_{3}, R_{3}, \cdots \). As \( n \) approaches infinity, to which point does \( P_{n} \) converge?

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