
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 a "position" as a point $(x, y)$ on the Cartesian plane, where $x$ and $y$ are positive integers not exceeding 20. Initially, all 400 positions are empty. Two players, A and B, take turns placing stones, starting with A. Each time it is A's turn, he places a new red stone on an empty position such that any two red stones are not at a distance of $\sqrt{5}$. Each time it is B's turn, he places a new blue stone on any empty position (the distance between the position of a blue stone and other stones can be any value). The game continues until one player can no longer place a stone. Determine the largest integer $K$ such that no matter how B places the blue stones, A can always guarantee placing at least $K$ red stones.

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