
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 side \( AB \) of an equilateral triangle \( \triangle ABC \). From \( P_{1} \), draw a perpendicular to \( BC \), with the foot of the perpendicular being \( Q_{1} \). From \( Q_{1} \), draw a perpendicular to \( CA \), with the foot of the perpendicular being \( R_{1} \). From \( R_{1} \), draw a perpendicular to \( AB \), with the foot of the perpendicular being \( P_{2} \). Repeat the same process starting from \( P_{2} \) to successively get points \( Q_{2} \), \( R_{2} \), \( P_{3} \), \( Q_{3} \), \( R_{3} \), and so on. As \( n \rightarrow \infty \), to which point does \( P_{n} \) converge?

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