
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
Alex and Bobette are playing on a $20 \times 20$ grid where the cells are square and have a side length of 1. The distance between two cells is the distance between their centers. They take turns playing as follows: Alex places a red stone on a cell, ensuring that the distance between any two cells with red stones is never $\sqrt{5}$, then Bobette places a blue stone on the grid without any restriction. The game ends when one of them cannot place a stone anymore. Find the largest $K$ such that Alex can always place at least $K$ stones, no matter how Bobette responds.

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