
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  $k$  be a positive integer. Scrooge McDuck owns  $k$  gold coins. He also owns infinitely many boxes  $B_1, B_2, B_3, \ldots$  Initially, bow  $B_1$  contains one coin, and the  $k-1$  other coins are on McDuck's table, outside of every box.
Then, Scrooge McDuck allows himself to do the following kind of operations, as many times as he likes:
- if two consecutive boxes  $B_i$  and  $B_{i+1}$  both contain a coin, McDuck can remove the coin contained in box  $B_{i+1}$  and put it on his table;
- if a box  $B_i$  contains a coin, the box  $B_{i+1}$  is empty, and McDuck still has at least one coin on his table, he can take such a coin and put it in box  $B_{i+1}$ .
As a function of  $k$ , which are the integers  $n$  for which Scrooge McDuck can put a coin in box  $B_n$ ?

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