
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
Given three sequences $\{F_n\}$, $\{k_n\}$, $\{r_n\}$ satisfying: $F_1=F_2=1$, $F_{n+2}=F_{n+1}+F_n$ ($n\in\mathbb{N}^*$), $r_n=F_n-3k_n$, $k_n\in\mathbb{N}$, $0\leq r_n<3$, calculate the sum $r_1+r_3+r_5+\ldots+r_{2011}$.

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