
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  $C$  be the set of all 100-digit numbers consisting of only the digits  $1$  and  $2$ . Given a number in  $C$ , we may transform the number by considering any  $10$  consecutive digits  $x_0x_1x_2 \dots x_9$  and transform it into  $x_5x_6\dots x_9x_0x_1\dots x_4$ . We say that two numbers in  $C$  are similar if one of them can be reached from the other by performing finitely many such transformations. Let  $D$  be a subset of  $C$  such that any two numbers in  $D$  are not similar. Determine the maximum possible size of  $D$ .

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