
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
For any two convex polygons  $P_1$  and  $P_2$  with mutually distinct vertices, denote by  $f(P_1, P_2)$  the total number of their vertices that lie on a side of the other polygon. For each positive integer  $n \ge 4$ , determine
\[ \max \{ f(P_1, P_2) ~ | ~ P_1 ~ \text{and} ~ P_2 ~ \text{are convex} ~ n \text{-gons} \}. \]
(We say that a polygon is convex if all its internal angles are strictly less than  $180^\circ$ .)

*Josef Tkadlec (Czech Republic)*

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