
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
A  $k \times k$  array contains each of the numbers  $1, 2, \dots, m$  exactly once, with the remaining entries all zero. Suppose that all the row sums and column sums are equal. What is the smallest possible value of  $m$  if  $k = 3^n$  ( $n \in \mathbb{N}^+$ )?

*Attila Sztranyák and Péter Erben*

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