
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 \( AB = 6 \), \( BC = 8 \), and \( AC = 10 \). 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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