
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
There are 2019 bags, each bag containing 2019 beads with a total weight of 1 kilogram. The beads in each bag are numbered \(1, 2, \cdots, 2019\). A selection is considered proper if and only if beads with different numbers are selected from different bags (at most one bead from each bag), and the total weight is at least 1 kilogram. Find the maximum value of \(k\) such that there always exist at least \(k\) different proper selections.

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