
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
Alice and Bob play on a $20 \times 20$ grid. Initially, all the cells are empty. Alice starts and the two players take turns placing stones on unoccupied cells. On her turn, Alice places a red stone on an empty cell that is not at a distance of $\sqrt{5}$ from any other cell containing a red stone. On his turn, Bob places a blue stone on an unoccupied cell. The game ends when a player can no longer place a stone. Determine the largest $K$ such that Alice can ensure to place at least $K$ red stones regardless of how Bob places his stones.

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