
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
Let the function  $f$  be defined on the set  $\mathbb{N}$  such that $\text{(i)}\ \ \quad f(1)=1$  $\text{(ii)}\ \quad f(2n+1)=f(2n)+1$  $\text{(iii)}\quad f(2n)=3f(n)$ Determine the set of values taken  $f$ .

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