
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
Triangle \( ABC \) is an equilateral triangle with side length 1. Let \( X_0, X_1, \ldots \) be an infinite sequence of points such that the following conditions hold:
- \( X_0 \) is the center of \( ABC \).
- For all \( i \geq 0 \), \( X_{2i+1} \) lies on segment \( AB \) and \( X_{2i+2} \) lies on segment \( AC \).
- For all \( i \geq 0 \), \(\measuredangle X_i X_{i+1} X_{i+2}=90^\circ\).
- For all \( i \geq 1 \), \( X_{i+2} \) lies in triangle \( A X_i X_{i+1} \).

Find the maximum possible value of \(\sum_{i=0}^{\infty} |X_i X_{i+1}|\), where \(|PQ|\) is the length of line segment \( PQ \).

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