
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 \( a_1, a_2, \ldots, a_{2n}, b_1, b_2, \ldots, b_{2n} \) be \( 4n \) non-negative real numbers, satisfying

\[ a_1 + a_2 + \cdots + a_{2n} = b_1 + b_2 + \cdots + b_{2n} > 0, \]

and for any \( i = 1, 2, \ldots, 2n \), we have \( a_i a_{i+2} \geq b_i + b_{i+1} \), where \( a_{2n+1} = a_1, a_{2n+2} = a_2, b_{2n+1} = b_1 \). Find the minimum value of \( a_1 + a_2 + \cdots + a_{2n} \).

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