
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
Xiao Hui, a collector, purchased a collectible item M for $$200,000$$ yuan at the beginning of the first year. Due to the influence of the collectibles market, at the beginning of the second and third year, the value of M is half that of the beginning of the previous year. Starting from the fourth year, at the beginning of each year, the value of M increases by $$40,000$$ yuan.  
(Ⅰ) Determine in which year the value of M exceeds its original purchase price.  
(Ⅱ) Let $$T_n$$ (n∈ℕ*) represent the average value of the collectible M over the first n years. Find the minimum value of $$T_n$$.

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