
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
Define a function $g(n)$ with the form:
\[g(n) = \frac{5 + 4\sqrt{2}}{2}\left(\sqrt{2}\right)^n - \frac{5 - 4\sqrt{2}}{2}\left(-\sqrt{2}\right)^n.\]
Calculate the expression for $g(n+1) - g(n-1)$ in terms of $g(n)$.
A) $\sqrt{2} g(n)$  
B) $\frac{\sqrt{2}}{2} g(n)$  
C) $\frac{1}{2} g(n)$  
D) $2 g(n)$

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