
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
There is a $2015 \times 2015$ grid board. Dima places detectors in $k$ cells. Then Kolya places a grid-based ship in the shape of a $1500 \times 1500$ square on the board. A detector in a cell reports to Dima whether that cell is covered by the ship or not. What is the smallest $k$ for which Dima can place the detectors to guarantee determining the position of the ship?

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