
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
The decomposition rate of a certain type of garbage is given by the function v = ab^t, where v is the decomposition rate and t is the time in months. Given that the decomposition rate is 10% after 12 months and 20% after 24 months, determine the time it will take for this type of garbage to completely decompose.

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