
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 positive integers  $k,n$  with  $k\leq n$ , we say that a  $k$ -tuple  $\left(a_1,a_2,\ldots,a_k\right)$  of positive integers is *tasty* if


- there exists a  $k$ -element subset  $S$  of  $[n]$  and a bijection  $f:[k]\to S$  with  $a_x\leq f\left(x\right)$  for each  $x\in [k]$ ,
- $a_x=a_y$  for some distinct  $x,y\in [k]$ , and
- $a_i\leq a_j$  for any  $i < j$ .

For some positive integer  $n$ , there are more than  $2018$  tasty tuples as  $k$  ranges through  $2,3,\ldots,n$ . Compute the least possible number of tasty tuples there can be.

Note: For a positive integer  $m$ ,  $[m]$  is taken to denote the set  $\left\{1,2,\ldots,m\right\}$ .

*Proposed by Vincent Huang and Tristan Shin*

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