
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
Determine all real functions \( f(x) \) that are defined and continuous on the interval \((-1, 1)\) and that satisfy the functional equation
\[ f(x+y) = \frac{f(x) + f(y)}{1 - f(x) f(y)} \quad \left( x, y, x+y \in (-1,1) \right). \]

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