
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
Define "position" as a point $(x, y)$ on the Cartesian coordinate plane, where both $x$ and $y$ are positive integers not exceeding 20. Initially, all 400 positions are empty. Two players, A and B, take turns placing stones, with A going first. Each time it is A's turn, A places a new red stone on an empty position, ensuring that the distance between any two red stones is never equal to $\sqrt{5}$. Each time it is B's turn, B places a new blue stone on any empty position (the distance between a blue stone and other stones can be any value). The game continues until neither player can place any more stones. Determine the largest integer $K$ such that no matter how B places the blue stones, A can always ensure that at least $K$ red stones are placed.

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