
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 real numbers \( a_1, a_2, \ldots, a_n \in [0, 2] \) with the condition that \( a_{n+1} = a_1 \), find the maximum value of the expression \(\frac{\sum_{i=1}^{n} a_{1}^{2} a_{i+1} + 8n}{\sum_{i=1}^{n} a_{i}^{2}}\).

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