
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  $k$ ,  $m$  and  $n$  be integers such that  $1<n \leq m-1 \leq k$ . Find maximum size of subset  $S$  of set  $\{1,2,...,k\}$  such that sum of any  $n$  different elements from  $S$  is not: $a)$  equal to  $m$ , $b)$  exceeding  $m$  

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