
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 a given positive integer $k$, find the smallest positive integer $N$ such that there exists a set consisting of $2k + 1$ distinct positive integers whose total sum is greater than $N$, but the sum of any $k$-element subset is at most $\frac{N}{2}$.

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