
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 the function \( g(n) \) as follows:
\[
g(n) = \dfrac{2+\sqrt{2}}{4}\left(\dfrac{1+\sqrt{2}}{2}\right)^n + \dfrac{2-\sqrt{2}}{4}\left(\dfrac{1-\sqrt{2}}{2}\right)^n.
\]
Determine \( g(n+1) - g(n-1) \) and express it in terms of \( g(n) \).
A) \( \dfrac{1}{2} g(n) \)
B) \( g(n) \)
C) \( 2g(n) \)
D) \( \dfrac{\sqrt{2}}{2} g(n) \)
E) \( \dfrac{1}{2}(g^2(n)-1) \)

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