
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 \( k, a_2, a_3 \) and \( k, b_2, b_3 \) be nonconstant geometric sequences with different common ratios. Given that \( a_3 - b_3 = 4(a_2 - b_2) \), find the sum of the common ratios of the two sequences.

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