
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
Given integers \(m\) and \(n\) such that \(2 \leq m < n\) and \(\gcd(m, n) = 1\), find the smallest integer \(k\) such that for any \(m\)-element subset \(I\) of the set \(\{1, 2, \ldots, n\}\), if \(\sum_{i \in I} i > k\), then there exist \(n\) real numbers \(a_1 \leq a_2 \leq \cdots \leq a_n\) such that
$$
\frac{1}{m} \sum_{i \in I} a_i > \frac{1}{n} \sum_{i=1}^{n} a_i.
$$

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