
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
In the set  $ S_n \equal{} \{1, 2,\ldots ,n\}$  a new multiplication  $ a*b$  is defined with the following properties:

**(i)**  $ c \equal{} a * b$  is in  $ S_n$  for any  $ a \in S_n, b \in S_n.$ 
**(ii)** If the ordinary product  $ a \cdot b$  is less than or equal to  $ n,$  then  $ a*b \equal{} a \cdot b.$ 
**(iii)** The ordinary rules of multiplication hold for  $ *,$  i.e.:

**(1)**  $ a * b \equal{} b * a$  (commutativity)
**(2)**  $ (a * b) * c \equal{} a * (b * c)$  (associativity)
**(3)** If  $ a * b \equal{} a * c$  then  $ b \equal{} c$  (cancellation law).

Find a suitable multiplication table for the new product for  $ n \equal{} 11$  and  $ n \equal{} 12.$ 

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