
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  $S={1,2,\ldots,100}$ . Consider a partition of  $S$  into  $S_1,S_2,\ldots,S_n$  for some  $n$ , i.e.  $S_i$  are nonempty, pairwise disjoint and  $\displaystyle S=\bigcup_{i=1}^n S_i$ . Let  $a_i$  be the average of elements of the set  $S_i$ . Define the score of this partition by 
\[\dfrac{a_1+a_2+\ldots+a_n}{n}.\]

Among all  $n$  and partitions of  $S$ , determine the minimum possible score.

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