
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
n and r are positive integers. Find the smallest k for which we can construct r subsets \( A_1, A_2, \ldots, A_r \) of \( \{0, 1, 2, \ldots, n-1\} \) each with k elements such that each integer \( 0 \le m < n \) can be written as a sum of one element from each of the r subsets.

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