
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

Functions \( f, g: \mathbb{R} \rightarrow \mathbb{R} \) are non-constant and satisfy the following identities for all \( x, y \in \mathbb{R} \):

\[
\begin{aligned}
& f(x+y)=f(x) g(y)+g(x) f(y), \\
& g(x+y)=g(x) g(y)-f(x) f(y).
\end{aligned}
\]

Find all possible values of \( f(0) \) and \( g(0) \).

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