
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
Given that the decomposition rate $v$ of a certain type of garbage approximately satisfies the relationship $v=a\cdot b^{t}$, where $a$ and $b$ are positive constants, and the decomposition rate is $5\%$ after $6$ months and $10\%$ after $12$ months, calculate the time it takes for this type of garbage to completely decompose.

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