
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
In an exotic country, the National Bank issues coins that can take any value in the interval  $[0, 1]$ . Find the smallest constant  $c > 0$  such that the following holds, no matter the situation in that country:

*Any citizen of the exotic country that has a finite number of coins, with a total value of no more than  $1000$ , can split those coins into  $100$  boxes, such that the total value inside each box is at most  $c$ .*

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