
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
Consider a convex polygon  $A_{1}A_{2}\ldots A_{n}$  and a point  $M$  inside it. The lines  $A_{i}M$  intersect the perimeter of the polygon second time in the points  $B_{i}$ . The polygon is called balanced if all sides of the polygon contain exactly one of points  $B_{i}$  (strictly inside). Find all balanced polygons.

[Note: The problem originally asked for which  $n$  all convex polygons of  $n$  sides are balanced. A misunderstanding made this version of the problem appear at the contest]

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