
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
For each integer \( n \geqslant 2 \), determine the maximum value of the sum \( a_{0} + a_{1} + a_{2} + \cdots + a_{n} \) given the conditions \( a_{0} = 1 \) and \( a_{i} \geqslant a_{i+1} + a_{i+2} \) for \( i = 0, 1, \cdots, n-2 \), where \( a_{0}, a_{1}, \cdots, a_{n} \) are non-negative.

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