
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 \( n \geq 3 \), let \( a_{1}, a_{2}, \cdots, a_{2n} \) and \( b_{1}, b_{2}, \cdots, b_{2n} \) be \( 4n \) non-negative real numbers that satisfy \( a_{1}+a_{2}+\cdots+a_{2n} = b_{1}+b_{2}+\cdots+b_{2n} > 0 \), and for any \( i=1, 2, \cdots, 2n \), we have \( a_{i}a_{i+2} \geq b_{i} + b_{i+1} \) (here \( a_{2n+1}=a_{1}, a_{2n+2}=a_{2}, b_{2n+1}=b_{1} \)).

Find the minimum value of \( a_{1} + a_{2} + \cdots + a_{2n} \).

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