
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
Let \( a_{1}, a_{2}, \cdots, a_{20} \) be 20 distinct positive integers, and let the set \( \left\{ a_{i} + a_{j} \mid 1 \leq i \leq j \leq 20 \right\} \) contain 201 distinct elements. Find the minimum possible number of distinct elements in the set \( \left\{ |a_{i} - a_{j}| \mid 1 \leq i < j \leq 20 \right\} \).

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