
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 that \( m \) and \( n \) are integers greater than 7, consider a rectangular grid formed by \( m \times n \) points. We color \( k \) of these points red such that no right-angled triangle with two sides parallel to the sides of the rectangle has all three vertices colored red. Find the maximum value of \( k \).

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