
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
Given \( n \) real numbers, satisfying \( \left|x_{i}\right| < 1 \) for \( i = 1, \ldots, n \), and \( \left| x_1 \right| + \left| x_2 \right| + \cdots + \left| x_n \right| \geq 19 + \left| x_1 + \cdots + x_n \right| \), find the smallest possible value of \( n \).

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