
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 \( k \geq 1 \) be an integer. Find the smallest integer \( N \) for which there exist \( 2k + 1 \) distinct integers \( a_{1}, \ldots, a_{2k+1} \geq 1 \) satisfying:

- The sum of the \( a_{i} \)'s is at least \( N \).

- The sum of any \( k \) distinct integers chosen among the \( a_{i} \)'s is always less than or equal to \( \frac{N}{2} \).

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