
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
Let  $m$  and  $n$  be positive integers. Some squares of an  $m \times n$  board are coloured red. A sequence  $a_1, a_2, \ldots , a_{2r}$  of  $2r \ge 4$  pairwise distinct red squares is called a *bishop circuit* if for every  $k \in \{1, \ldots , 2r \}$ , the squares  $a_k$  and  $a_{k+1}$  lie on a diagonal, but the squares  $a_k$  and  $a_{k+2}$  do not lie on a diagonal (here  $a_{2r+1}=a_1$  and  $a_{2r+2}=a_2$ ).
In terms of  $m$  and  $n$ , determine the maximum possible number of red squares on an  $m \times n$  board without a bishop circuit.
(*Remark.* Two squares lie on a diagonal if the line passing through their centres intersects the sides of the board at an angle of  $45^\circ$ .)

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