
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
Given the sets  $R_{mn} = \{ (x,y) \mid x=0,1,\dots,m; y=0,1,\dots,n \}$ , consider functions  $f:R_{mn}\to \{-1,0,1\}$  with the following property: for each quadruple of points  $A_1,A_2,A_3,A_4\in R_{mn}$  which form a square with side length  $0<s<3$ , we have  $$ f(A_1)+f(A_2)+f(A_3)+f(A_4)=0. $$  For each pair  $(m,n)$  of positive integers, determine  $F(m,n)$ , the number of such functions  $f$  on  $R_{mn}$ .

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