
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
A rectangular region is bounded by the graphs of the equations \(y = 2a\), \(y = -b\), \(x = -2c\), and \(x = d\), where \(a, b, c,\) and \(d\) are all positive numbers. There is also a line \(y = x + a\) intersecting the rectangle. Determine the area of the rectangle that is below the line \(y = x + a\).
A) \((2a + b)(d + 2c) - \frac{1}{2} (d + 2c + b + a)(a + b + 2c)\) \\
B) \((2a + b)(d + 2c) + \frac{1}{2} (d + 2c + b + a)(a + b + 2c)\) \\
C) \(\frac{1}{2} (d + 2c + b + a)(a + b + 2c)\) \\
D) \((2a + b)(d + 2c)\)

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