
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 natural numbers \( a \) and \( b \), where \( a < b < 2a \), some cells are marked on a grid such that in each grid rectangle of size \( a \times b \) or \( b \times a \), at least one cell is marked. What is the maximum \( \alpha \) for which it can be asserted that for any natural number \( N \), there exists a grid square of size \( N \times N \) in which at least \( \alpha N^{2} \) cells are marked?

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