
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 \( a \) and \( b \) as specific positive integers that satisfy \( a < b < 2a \). A rectangle in the coordinate plane, with vertices at grid points and sides parallel to the coordinate axes, is called a "standard rectangle." A standard rectangle with an area of 1 is called a "unit square." Some unit squares in the coordinate plane are colored red. It is known that any \( a \times b \) or \( b \times a \) standard rectangle includes at least one red unit square. Find the maximum constant \( \alpha \) such that for any positive integer \( N \), there exists an \( N \times N \) standard rectangle containing at least \( \alpha N^{2} \) red unit squares.

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