
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
Define an infinite sequence of 0s and 1s as follows: Starting from 0, repeatedly change each 0 to 001 and each 1 to 0.
(a) Is this sequence periodic?
(b) What is the 1000th digit in this sequence?
(c) At what position does the 10000th occurrence of the digit 1 appear?
(d) Find the formulas for the positions where the digit 1 appears (e.g., 3, 6, 10, 13, ...) and the positions where the digit 0 appears.

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