
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
Given a point  $p$  and a line segment  $l$ , let  $d(p,l)$  be the distance between them.  Let  $A$ ,  $B$ , and  $C$  be points in the plane such that  $AB=6$ ,  $BC=8$ ,  $AC=10$ .  What is the area of the region in the  $(x,y)$ -plane formed by the ordered pairs  $(x,y)$  such that there exists a point  $P$  inside triangle  $ABC$  with  $d(P,AB)+x=d(P,BC)+y=d(P,AC)?$ 

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