
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.  Determine the least positive integer  $n$ , with  $n\geq k+1$ , for which the game below can be played indefinitely:

Consider  $n$  boxes, labelled  $b_1,b_2,...,b_n$ . For each index  $i$ , box  $b_i$  contains exactly  $i$  coins. At each step, the following three substeps are performed in order:**(1)** Choose  $k+1$  boxes;**(2)** Of these  $k+1$  boxes, choose  $k$  and remove at least half of the coins from each, and add  to the remaining box, if labelled  $b_i$ , a number of  $i$  coins.**(3)** If one of the boxes is left empty, the game ends; otherwise, go to the next step.

*Proposed by Demetres Christofides, Cyprus*

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