
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 \( x_1, x_2, \ldots, x_n \) be real numbers such that each \( |x_i| < 1 \) and \( |x_i| \neq |x_j| \) for \( i \neq j \). Determine the smallest possible value of \( n \) for which the equation
\[
|x_1| + |x_2| + \dots + |x_n| = 25 + |x_1 + x_2 + \dots + x_n|
\] 
holds.

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