
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
Santa Claus has many identical dials in the form of regular 12-sided polygons, with numbers from 1 to 12 printed on them. He stacks these dials one on top of the other (face-up). The vertices of the dials are aligned, but the numbers at the aligned vertices do not necessarily match. The Christmas tree will light up as soon as the sums of the numbers in all 12 columns have the same remainder when divided by 12. How many dials might be in the stack at that moment?

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