
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 any given integers  $m,n$  such that  $2\leq m<n$  and  $(m,n)=1$ . Determine the smallest positive integer  $k$  satisfying the following condition: for any  $m$ -element subset  $I$  of  $\{1,2,\cdots,n\}$  if  $\sum_{i\in I}i> k$ , then there exists a sequence of  $n$  real numbers  $a_1\leq a_2 \leq \cdots \leq a_n$  such that  $$ \frac1m\sum_{i\in I} a_i>\frac1n\sum_{i=1}^na_i $$  

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