
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
Let  $S$  be the set of 81 points  $(x, y)$  such that  $x$  and  $y$  are integers from  $-4$  through  $4$ .  Let  $A$ ,  $B$ , and  $C$  be random points chosen independently from  $S$ , with each of the 81 points being equally likely.  (The points  $A$ ,  $B$ , and  $C$  do not have to be different.)  Let  $K$  be the area of the (possibly degenerate) triangle  $ABC$ .  What is the expected value (average value) of  $K^2$  ?

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