
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
Philena and Nathan are playing a game. Nathan secretly chooses an ordered pair \((x, y)\) of positive integers such that \(x \leq 20\) and \(y \leq 23\). Philena knows that Nathan's pair must satisfy \(x \leq 20\) and \(y \leq 23\). The game proceeds in rounds where, in each round, Philena chooses an ordered pair \((a, b)\) of positive integers and tells it to Nathan. Nathan responds with YES if \(x \leq a\) and \(y \leq b\), and NO otherwise. Find, with proof, the smallest positive integer \(N\) for which Philena can be certain of Nathan's pair after at most \(N\) rounds.

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