
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
We are given an infinite sequence of  $ 1$ 's and  $ 2$ 's with the following properties: 

(1) The first element of the sequence is  $ 1$ . 

(2) There are no two consecutive  $ 2$ 's or three consecutive  $ 1$ 's. 

(3) If we replace consecutive  $ 1$ 's by a single  $ 2$ , leave the single  $ 1$ 's alone, and delete the original  $ 2$ 's, then we recover the original sequence. 

How many  $ 2$ 's are there among the first  $ n$  elements of the sequence? 

*P. P. Palfy*

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