
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
Let $f(x) = e^x - e^{-x}$ and $g(x) = e^x + e^{-x}$.

(1) Find the value of $[f(x)]^2 - [g(x)]^2$;  
(2) If $f(x) \cdot f(y) = 4$ and $g(x) \cdot g(y) = 8$, find the value of $\frac{g(x+y)}{g(x-y)}$.

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