
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
Find the largest positive integer \( n \) such that there exist 8 integers \( x_{1}, x_{2}, x_{3}, x_{4} \) and \( y_{1}, y_{2}, y_{3}, y_{4} \) satisfying:
\[ \{0,1, \cdots, n\} \subseteq \left\{ |x_{i} - x_{j}| \mid 1 \leq i < j \leq 4 \right\} \cup \left\{ |y_{i} - y_{j}| \mid 1 \leq i < j \leq 4 \right\}. \]

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