
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
A field is made of  $2017 \times 2017$  unit squares. Luffy has  $k$  gold detectors, which he places on some of the unit squares, then he leaves the area. Sanji then chooses a  $1500 \times 1500$  area, then buries a gold coin on each unit square in this area and none other. When Luffy returns, a gold detector beeps if and only if there is a gold coin buried underneath the unit square it's on. It turns out that by an appropriate placement, Luffy will always be able to determine the  $1500 \times 1500$  area containing the gold coins by observing the detectors, no matter how Sanji places the gold coins. Determine the minimum value of  $k$  in which this is possible.

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