
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
Yakob and Baptiste play on a $20 \times 20$ grid where the cells are square with a side length of 1. The distance between two cells is the distance between their centers. They take turns as follows: Yakob places a red stone on a cell, ensuring that the distance between two cells with red stones is never $\sqrt{5}$, and Baptiste places a blue stone on the grid without any restrictions. The game stops when one of them can no longer place a stone. Find the largest $K$ such that Yakob can always place at least $K$ stones, regardless of Baptiste's responses.

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