
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 \( M = \{1, 2, \cdots, 10\} \) and let \( T \) be a collection of certain two-element subsets of \( M \), such that for any two different elements \(\{a, b\} \) and \(\{x, y\} \) in \( T \), the condition \( 11 \nmid (ax + by)(ay + bx) \) is satisfied. 

Find the maximum number of elements in \( T \).

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