
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
A calculator can accept a pair of integers for conversion, and it changes the pair of integers (a, b) by pressing a button according to the following rules: when the sum of the two numbers in the original pair of integers (a, b), a+b, is divided by 4, the remainders 0, 1, 2, 3 correspond to the outputs (a+1, b), (a, b+1), (a-1, b), (a, b-1) respectively.  
(1) If the original pair of integers is (1, 12), what is the pair of integers (x, y) after 6 transformations, and what is the value of x+y?  
(2) If the pair of integers after 6 transformations is (1, 17), what is the value of the original pair of integers?

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