
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 \( a, b, c, d \) be positive real numbers such that \( a < b \), \( c < d \), and \( bc > ad \). A triangle has side lengths \(\sqrt{a^2 + c^2}\), \(\sqrt{b^2 + d^2}\), and \(\sqrt{(b-a)^2 + (d-c)^2}\). Find the area of this triangle.

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