
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 with 0, repeatedly replace each 0 with 001, and each 1 with 0.

(a) Is this sequence periodic?
(b) What is the 1000th digit in this sequence?
(c) At which position does the 10,000th '1' appear in this sequence?
(d) Find the formula for the positions of digit '1' (3, 6, 10, 13, ...) and the formula for the positions of digit '0'.

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