
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
Consider the points  $A(0,12)$ ,  $B(10,9)$ ,  $C(8,0)$ , and  $D(-4,7)$ . There is a unique square  $S$  such that each of the four points is on a different side of  $S$ . Let  $K$  be the area of  $S$ . Find the remainder when  $10K$  is divided by  $1000$ .

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