
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
Willy Wonka has  $n$  distinguishable pieces of candy that he wants to split into groups. If the number of ways for him to do this is  $p(n)$ , then we have 
        \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\hline
             $n$  &  $1$  &  $2$  &  $3$  &  $4$  &  $5$  &  $6$  &  $7$  &  $8$  &  $9$  &  $10$  \hline
             $p(n)$  &  $1$  &  $2$  &  $5$  &  $15$  &  $52$  &  $203$  &  $877$  &  $4140$  &  $21147$  &  $115975$  \hline
        \end{tabular}
    Define a *splitting* of the  $n$  distinguishable pieces of candy to be a way of splitting them into groups. If Willy Wonka has  $8$  candies, what is the sum of the number of groups over all splittings he can use?

*2020 CCA Math Bonanza Lightning Round #3.4*

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