
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
Consider the sequence \( a_{1}, a_{2}, \cdots, a_{20} \) of 20 distinct positive integers such that the set \( \left\{a_{i}+a_{j} \mid 1 \leqslant i \leqslant j \leqslant 20\} \) contains 201 distinct elements. Find the minimum possible number of distinct elements in the set \( \left\{\left|a_{i}-a_{j}\right| \mid 1 \leqslant i < j \leqslant 20\right\} \).

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