
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 \( c > 0 \) be a given positive real number and \(\mathbb{R}_{>0}\) be the set of all positive real numbers. Find all functions \( f: \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0} \) such that

\[ f((c+1)x + f(y)) = f(x + 2y) + 2cx \]

for all \( x, y \in \mathbb{R}_{>0} \).

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