
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 two sequences $f(n)$ and $g(n)$ as follows:
\[ f(n) = \frac{3+\sqrt{3}}{6} \left(\frac{1+\sqrt{3}}{2}\right)^n + \frac{3-\sqrt{3}}{6} \left(\frac{1-\sqrt{3}}{2}\right)^n \]
\[ g(n) = \frac{2+\sqrt{2}}{4} \left(\frac{1+\sqrt{2}}{2}\right)^n + \frac{2-\sqrt{2}}{4} \left(\frac{1-\sqrt{2}}{2}\right)^n \]
Determine the value of $g(n+1) + g(n-1)$ in terms of $f(n)$.
A) $f(n)$
B) $f(n) + 1$
C) $2f(n)$
D) $4f(n)$

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