
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
Richard has an infinite row of empty boxes labeled  $1, 2, 3, \ldots$  and an infinite supply of balls. Each minute, Richard finds the smallest positive integer  $k$  such that box  $k$  is empty. Then, Richard puts a ball into box  $k$ , and if  $k \geq 3$ , he removes one ball from each of boxes  $1,2,\ldots,k-2$ . Find the smallest positive integer  $n$  such that after  $n$  minutes, both boxes  $9$  and  $10$  have at least one ball in them.

*Proposed by **vvluo** & **richy***

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