
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
A subset of the positive integers  $S$  is said to be a \emph{configuration} if 200  $\notin S$  and for all nonnegative integers  $x$ ,  $x \in S$  if and only if both 2 $x\in S$  and  $\left \lfloor{\frac{x}{2}}\right \rfloor\in S$ . Let the number of subsets of  $\{1, 2, 3, \dots, 130\}$  that are equal to the intersection of  $\{1, 2, 3, \dots, 130\}$  with some configuration  $S$  equal  $k$ . Compute the remainder when  $k$  is divided by 1810.  

*Proposed Hari Desikan (HariDesikan)*

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