
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 \( m \) be any natural number. As a function of \( m \), determine the smallest natural number \( k \) for which the following applies: If \( \{m, m+1, \ldots, k\} = A \cup B \) is any decomposition into two sets \( A \) and \( B \), then \( A \) or \( B \) contains three elements \( a, b, c \) (which do not necessarily have to be different) with \( a^b = c \).

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