
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 \( f: (0, +\infty) \rightarrow (0, +\infty) \) that satisfy the following condition for any positive real numbers \( x \) and \( y \):
\[ f(f(x) + y) = f(f(x)) + 2y f(x) - f(y) + 2y^2 + 1. \]

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