
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 set of 2-element subsets of  $M$ . For any two different elements  $\{a,b\}, \{x,y\}$  in  $T$ , the integer  $(ax+by)(ay+bx)$  is not divisible by 11. Find the maximum size of  $T$ .

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