
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
Find all integers  $n$  greater than or equal to  $4$  that satisfy the following conditions:


- Take an arbitrary convex  $n$ -gon  $P$  on the coordinate plane whose vertices are lattice points (points whose coordinates are both integers). There are a total of  $n$  triangles that share two sides with  $P$ . Let  $S_1, S_2, \ldots, S_n$  be their areas, and let  $S$  be the area of  $P$ . Then, the greatest common divisor of  $2S_1, 2S_2, \ldots, 2S_n$  divides the integer  $2S$ .


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