
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
Let \( x_{1}, \ldots, x_{100} \) be positive real numbers such that \( x_{i} + x_{i+1} + x_{i+2} \leq 1 \) for all \( i \leq 100 \) (with the convention that \( x_{101} = x_{1} \) and \( x_{102} = x_{2} \)). Determine the maximum value of 

\[ 
S = \sum_{i=1}^{100} x_{i} x_{i+2} 
\]

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