
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
Suppose $x_{1}, x_{2}, \cdots, x_{n}$ and $a_{1}, a_{2}, \cdots, a_{n}$ are two sets of arbitrary real numbers that satisfy the following conditions: 
- $1^{\circ} . \sum_{i=1}^{n} x_{i}=0$
- $2^{\circ} . \sum_{i=1}^{n}\left|x_{i}\right|=1$
- $3^{\circ} . a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{n}$, where $n \geqslant 2$.

Find the minimum value of $A$ such that the inequality $\left|\sum_{i=1}^{n} a_{i} x_{i}\right| \leqslant A\left(a_{1}-a_{n}\right)$ holds.

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