
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
The functions \( f \) and \( g \) from \( \mathbb{R} \) to \( \mathbb{R} \) are not constant and satisfy the identities:
\[
\begin{cases}
f(x+y) = f(x) g(y) + g(x) f(y), \\
g(x+y) = g(x) g(y) - f(x) f(y),
\end{cases} \quad \text{for all } x, y \in \mathbb{R}.
\]
Find all possible values of \( f(0) \) and \( g(0) \).

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