
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  $n$  be a positive integer. Mariano divides a rectangle into  $n^2$  smaller rectangles by drawing  $n-1$  vertical lines and  $n-1$  horizontal lines, parallel to the sides of the larger rectangle. On every step, Emilio picks one of the smaller rectangles and Mariano tells him its area. Find the least positive integer  $k$  for which it is possible that Emilio can do  $k$  conveniently thought steps in such a way that with the received information, he can determine the area of each one of the  $n^2$  smaller rectangles.


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