
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  $ A = \{ 1, 2, 3, \cdots , 12 \} $ . Find the number of one-to-one function  $ f :A \to A $  satisfying following condition: for all  $ i \in A $ ,  $ f(i)-i $  is not a multiple of   $ 3 $ .

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