
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  $S$  and  $T$  be non-empty, finite sets of positive integers. We say that  $a\in\mathbb{N}$  is \emph{good} for  $b\in\mathbb{N}$  if  $a\geq\frac{b}{2}+7$ . We say that an ordered pair  $\left(a,b\right)\in S\times T$  is \emph{satisfiable} if  $a$  and  $b$  are good for each other.

A subset  $R$  of  $S$  is said to be \emph{unacceptable} if there are less than  $\left|R\right|$  elements  $b$  of  $T$  with the property that there exists  $a \in R$  such that  $\left(a,b\right)$  is satisfiable. If there are no unacceptable subsets of  $S$ , and  $S$  contains the elements  $14$ ,  $20$ ,  $16$ ,  $32$ ,  $23$ , and  $31$ , compute the smallest possible sum of elements of  $T$  given that  $\left|T\right|\geq20$ . 

*Proposed by Tristan Shin*

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