
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 non-negative real numbers \( a_{1} \geq a_{2} \geq \ldots \geq a_{n} \) and \( b_{1} \leq b_{2} \leq \ldots \leq b_{n} \) satisfying
$$
a_{1} a_{n}+a_{2} a_{n-1}+\ldots+a_{n} a_{1}=b_{1} b_{n}+b_{2} b_{n-1}+\ldots+b_{n} b_{1}=1,
$$
find the minimum value of
$$
S=\sum_{1 \leq i<j \leq n} a_{i} b_{j}.
$$

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