
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 the functions $f(x)$ and $g(x)$ are defined on $\mathbb{R}$, for any $x, y \in \mathbb{R}$, we have $f(x-y) = f(x)g(y) - g(x)f(y)$, and $f(x) \neq 0$. If $f(1) = f(2)$, then $g(-1) + g(1) = \ $.

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