
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
Infinitely many math beasts stand in a line, all six feet apart, wearing masks, and with clean hands. Grogg starts at the front of the line, holding  $n$  pieces of candy,  $ n \ge 1,$  and everyone else has none. He passes his candy to the beasts behind him, one piece each to the next  $n$  beasts in line. Then, Grogg leaves the line. The other beasts repeat this process: the beast in front, who has  $k$  pieces of candy, passes one piece each to the next  $k$  beasts in line, and then leaves the line. For some values of  $n,$  another beast, besides Grogg, temporarily holds all the candy. For which values of  $n$  does this occur?

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