
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

What is the maximum area that a rectangle can have if the coordinates of its vertices satisfy the equation \( |y-x| = (y+x+1)(5-x-y) \), and its sides are parallel to the lines \( y = x \) and \( y = -x \)? Give the square of the value of the maximum area found as the answer. (12 points)

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