
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 \(b\) such that \(a < b < 2a\). On a grid plane, some cells are marked in such a way that in each grid rectangle of size \(a \times b\) or \(b \times a\) there is at least one marked cell. What is the largest \(\alpha\) for which it can be asserted that for any natural number \(N\), there is a grid square of size \(N \times N\) with at least \(\alpha N^2\) marked cells?

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