
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  $n=2^{2018}$  and let  $S=\{1,2,\ldots,n\}$ . For subsets  $S_1,S_2,\ldots,S_n\subseteq S$ , we call an ordered pair  $(i,j)$  *murine* if and only if  $\{i,j\}$  is a subset of at least one of  $S_i, S_j$ . Then, a sequence of subsets  $(S_1,\ldots, S_n)$  of  $S$  is called *tasty* if and only if:

1) For all  $i$ ,  $i\in S_i$ . 
2) For all  $i$ ,  $\displaystyle\bigcup_{j\in S_i} S_j=S_i$ . 
3) There do not exist pairwise distinct integers  $a_1,a_2,\ldots,a_k$  with  $k\ge 3$  such that for each  $i$ ,  $(a_i, a_{i+1})$  is murine, where indices are taken modulo  $k$ . 
4)  $n$  divides  $1+|S_1|+|S_2|+\ldots+|S_n|$ .

Find the largest integer  $x$  such that  $2^x$  divides the number of tasty sequences  $(S_1,\ldots, S_n)$ .

[i]Proposed by Vincent Huang and Brandon Wang

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