
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
An infinite sequence of digits  $1$  and  $2$  is determined by the following two properties:

i) The sequence is built by writing, in some order, blocks  $12$  and blocks  $112.$ ii) If each block  $12$  is replaced by  $1$  and each block  $112$  by  $2$ , the same sequence is again obtained.

In which position is the hundredth digit  $1$ ? What is the thousandth digit of the sequence?


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