
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
The Lucas numbers  $L_n$  are defined recursively as follows:  $L_0=2,L_1=1,L_n=L_{n-1}+L_{n-2}$  for  $n\geq2$ . Let  $r=0.21347\dots$ , whose digits form the pattern of the Lucas numbers. When the numbers have multiple digits, they will "overlap," so  $r=0.2134830\dots$ , **not**  $0.213471118\dots$ . Express  $r$  as a rational number  $\frac{p}{q}$ , where  $p$  and  $q$  are relatively prime.

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