
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
Find all functions \( f: \mathbb{R}^{+} \to \mathbb{R}^{+} \), where \(\mathbb{R}^{+}\) denotes the positive reals, such that whenever \( a > b > c > d > 0 \) are real numbers with \( a d = b c \),
\[ f(a + d) + f(b - c) = f(a - d) + f(b + c) . \]

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