
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  $n$  and  $k$  be positive integers. Cathy is playing the following game. There are  $n$  marbles and  $k$  boxes, with the marbles labelled  $1$  to  $n$ . Initially, all marbles are placed inside one box. Each turn, Cathy chooses a box and then moves the marbles with the smallest label, say  $i$ , to either any empty box or the box containing marble  $i+1$ . Cathy wins if at any point there is a box containing only marble  $n$ .
Determine all pairs of integers  $(n,k)$  such that Cathy can win this game.

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