
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 \( x_{1}, x_{2}, \cdots, x_{n} \) and \( a_{1}, a_{2}, \cdots, a_{n} \) be two sets of arbitrary real numbers (where \( n \geqslant 2 \)) that satisfy the following conditions:
1. \( x_{1} + x_{2} + \cdots + x_{n} = 0 \)
2. \( \left| x_{1} \right| + \left| x_{2} \right| + \cdots + \left| x_{n} \right| = 1 \)
3. \( a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{n} \)

Determine the minimum value of the real number \( A \) such that the inequality \( \left| a_{1} x_{1} + a_{2} x_{2} + \cdots + a_{n} x_{n} \right| \leqslant A ( a_{1} - a_{n} ) \) holds, and provide a justification for this value.

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