
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 the set
\[ I = \left\{ \left( x_1, x_2, x_3, x_4 \right) \mid x_i \in \{1, 2, \cdots, 11\} \right\}, \]
determine the maximum size of the subset \(A\) of \(I\) such that for any
\[ \left( x_1, x_2, x_3, x_4 \right), \left( y_1, y_2, y_3, y_4 \right) \in A, \]
there exist \(i, j\) (where \(1 \leq i < j \leq 4\)) such that
\[ \left( x_i - x_j \right)\left( y_i - y_j \right) < 0. \]

Find the maximum value of \(|A|\), where \(|A|\) represents the number of elements in the set \(A\).

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