
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 positive integers \( a \) and \( b \) that satisfy \( a < b < 2a \). A rectangle on the coordinate plane with vertices at integer points and edges 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 on the coordinate plane are colored red. Given that any \( a \times b \) or \( b \times a \) standard rectangle contains 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}
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