
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 an integer \( n \geqslant 2 \). Let \( a_{1}, a_{2}, \cdots, a_{n} \) and \( b_{1}, b_{2}, \cdots, b_{n} \) be positive numbers that satisfy
\[ a_{1} + a_{2} + \cdots + a_{n} = b_{1} + b_{2} + \cdots + b_{n}, \]
and for any \( i, j \) ( \( 1 \leqslant i < j \leqslant n \)), it holds that \( a_{i}a_{j} \geqslant b_{i} + b_{j} \). Find the minimum value of \( a_{1} + a_{2} + \cdots + a_{n} \).

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