
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
(2007 Vietnam Mathematical Olympiad) Let \( b \) be a positive real number. Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that for any real numbers \( x \) and \( y \), the following holds:
\[ f(x+y) = f(x) \cdot 3^{b + f(y) - 1} + b^x \left(3^{b^3 + f(y) - 1}\right) - b^{x+y}. \]

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