
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
Given functions $f(x)$ and $g(x)$ are defined for all real numbers on $\mathbb{R}$, and for any real numbers $x$ and $y$, they satisfy $f(x-y) = f(x)g(y) - g(x)f(y)$. If $f(1) = f(2) \neq 0$, then find the value of $g(1) + g(-1)$.

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