
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 left bank of the river to the right, \( N \) natives crossed using a single boat, each time two rowing to the right and one returning alone. Initially, each native knew one joke, and each joke was unique. They did not share jokes on the banks, but in the boat, each would tell their companion all the jokes they knew at that moment. For every natural number \( k \), determine the smallest possible value of \( N \) such that it could happen that by the end, each native knew, apart from their own, at least \( k \) jokes.

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