
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
Finitely many polygons are placed in the plane. If for any two polygons of them, there exists a line through origin  $O$  that cuts them both, then these polygons are called "properly placed". Find the least  $m \in \mathbb{N}$ , such that for any group of properly placed polygons,  $m$  lines can drawn through  $O$  and every polygon is cut by at least one of these  $m$  lines.

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