
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
A certain product originally sold for $\text{25}$ yuan per unit with an annual sales volume of $\text{80,000}$ units.
$(\text{I})$ According to a market survey, if the price increases by $\text{1}$ yuan, the sales volume will decrease by $\text{2,000}$ units. What is the highest price per unit that can be set for the product without reducing the total sales revenue?
$(\text{II})$ In order to expand the influence of the product and increase next year's sales volume, the company decides to carry out a comprehensive technical reform and marketing strategy adjustment for the product, raising the price to $x$ yuan. The company plans to invest $\frac{1}{6}(x^{2}-600)$ ten thousand yuan as technical reform fees, $\text{50}$ ten thousand yuan as fixed promotion fees, and $\frac{1}{5}x$ ten thousand yuan as variable promotion fees. What is the minimum sales volume $a$ (in ten thousand units) that must be achieved next year to make it possible for next year's sales revenue to not be lower than the sum of the original revenue and the total investment? Also, find the per-unit price of the product under these conditions.

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