
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 \( ABC \) be a triangle such that \( AB = AC = 182 \) and \( BC = 140 \). Let \( X_1 \) lie on \( AC \) such that \( CX_1 = 130 \). Let the line through \( X_1 \) perpendicular to \( BX_1 \) at \( X_1 \) meet \( AB \) at \( X_2 \). Define \( X_2, X_3, \ldots \) as follows: for \( n \) odd and \( n \geq 1 \), let \( X_{n+1} \) be the intersection of \( AB \) with the perpendicular to \( X_{n-1} X_n \) through \( X_n \); for \( n \) even and \( n \geq 2 \), let \( X_{n+1} \) be the intersection of \( AC \) with the perpendicular to \( X_{n-1} X_n \) through \( X_n \). Find \( BX_1 + X_1X_2 + X_2X_3 + \ldots \)

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