
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 two different geometric sequences with nonconstant terms and different common ratios. If \[a_3 + 2a_2 = b_3 + 3b_2,\] what is the sum of the common ratios of these two sequences?

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