
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
At the beginning of this year, a fishing company purchased an ocean-going fishing vessel for 980,000 yuan. The annual fishing income is 500,000 yuan. It is known that the total expenses (including maintenance) for using the boat for $x$ years ($x \in \mathbb{N}^*$) amount to $2x^2+10x$ thousand yuan.

(1) After how many years will the boat start to be profitable (i.e., the total income exceeds total expenses, with this year being the first year)?
(2) After several years, there are two options for handling the boat:
① Sell the boat for 80,000 yuan when the total profit reaches its maximum value.
② Sell the boat for 260,000 yuan when the annual average profit reaches its maximum value.
Which option is more economical? Please explain your reasoning.

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