
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
Consider a geometric sequence where any term of the sequence is equal to $k$ times the sum of the next two terms. Given $k=3$ and the property holds with all terms positive, find the common ratio $r$.
A) $\frac{-1 - \sqrt{\frac{7}{3}}}{2}$
B) $\frac{-1 + \sqrt{\frac{7}{3}}}{2}$
C) $\sqrt{\frac{3}{7}}$
D) $\frac{1 + \sqrt{5}}{2}$

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