
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
If the functions \( f(x) \) and \( g(x) \) are defined for all real numbers, and they satisfy the equation \( f(x-y) = f(x) g(y) - g(x) f(y) \), with \( f(-2) = f(1) \neq 0 \), then find \( g(1) + g(-1) \).

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