
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 every integer \( n \geq 2 \), determine the minimum value that the sum \( a_{0}+a_{1}+\cdots+a_{n} \) can take for nonnegative numbers \( a_{0}, a_{1}, \ldots, a_{n} \) satisfying the condition
$$
a_{0}=1, \quad a_{i} \leq a_{i+1}+a_{i+2} \quad \text { for } i=0, \ldots, n-2.
$$

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