
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 an integer \( n \geq 2 \), let the integers \( a_{0}, a_{1}, a_{2}, \ldots, a_{n} \) satisfy \( 0 = a_{0} < a_{1} < \ldots < a_{n} = 2n - 1 \). Find the smallest possible number of elements in the set \(\left\{a_{i} + a_{j} \mid 0 \leq i \leq j \leq n\right\}\).

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