
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
For any positive integer  $n$ , let  $S_n$  denote the set of positive integers which cannot be written in the form  $an+2017b$  for nonnegative integers  $a$  and  $b$ . Let  $A_n$  denote the average of the elements of  $S_n$  if the cardinality of  $S_n$  is positive and finite, and  $0$  otherwise. Compute \[\left\lfloor\displaystyle\sum_{n=1}^{\infty}\frac{A_n}{2^n}\right\rfloor.\]

*Proposed by Tristan Shin*

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