
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
From the set \( \{1, 2, 3, \ldots, 999, 1000\} \), select \( k \) numbers. If among the selected numbers, there are always three numbers that can form the side lengths of a triangle, what is the smallest value of \( k \)? Explain why.

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