Find Next Digit

Can you find out the next digit in the following series ?

0, 0, 1, 3, 2, 6, 3, 9, 4, 12, 5, ?




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The following conversation takes place between them:
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What are the two numbers?

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Which word is least like the others? Third, fourth, fifth, sixth, seventh, eighth, ninth?

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Can you find the missing letter in the sequence?
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Katniss and Peeta are liars. They both lie on some specific days.

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Can you tell that one day of the week when both of them will say "Tomorrow, I will lie."?

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Challenging

There is a cryptic organization called Cicada 3301 that posts challenging puzzles online, possibly to recruit codebreakers and linguists.