
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 a non-constant sequence $\{a_{i}\}$ satisfying $a_{i+1}^{2} - a_{i} a_{i+1} + a_{i}^{2} = 0$, and $a_{i+1} \neq a_{i-1}$ for $i = 1, 2, \cdots, n$, with the conditions that for a given natural number $n$, $a_{1} = a_{n+1} = 1$, find the value of $\sum_{i=0}^{n-1} a_{i}$.

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