
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
Arnaldo and Bernaldo play a game where they alternate saying natural numbers, and the winner is the one who says  $0$ . In each turn except the first the possible moves are determined from the previous number  $n$  in the following way: write
\[n =\sum_{m\in O_n}2^m;\]
the valid numbers are the elements  $m$  of  $O_n$ . That way, for example, after Arnaldo says  $42= 2^5 + 2^3 + 2^1$ , Bernaldo must respond with  $5$ ,  $3$  or  $1$ .

We define the sets  $A,B\subset \mathbb{N}$  in the following way. We have  $n\in A$  iff Arnaldo, saying  $n$  in his first turn, has a winning strategy; analogously, we have  $n\in B$  iff Bernaldo has a winning strategy if Arnaldo says  $n$  during his first turn. This way,
\[A =\{0, 2, 8, 10,\cdots\}, B = \{1, 3, 4, 5, 6, 7, 9,\cdots\}\]
Define  $f:\mathbb{N}\to \mathbb{N}$  by  $f(n)=|A\cap \{0,1,\cdots,n-1\}|$ . For example,  $f(8) = 2$  and  $f(11)=4$ .
Find 
\[\lim_{n\to\infty}\frac{f(n)\log(n)^{2005}}{n}\]

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