
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 satisfies the function relationship $v=a\cdot b^{t}$ (where $a$ and $b$ are non-zero constants) with time $t$ (unit: months), and after $6$ months, the decomposition rate is $5\%$, and after $12$ months, the decomposition rate is $10\%$, determine how many months it will take for this garbage to completely decompose (decomposition rate of $100\%$).

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