
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 $\\(25\\)$ yuan per unit with an annual sales volume of $\\(8\\)$ ten thousand units.

\\((1)\\) According to market research, for every $\\(1\\)$ yuan increase in price, the sales volume will decrease by $\\(2000\\)$ units. What is the highest price per unit that can be set for the product without lowering the total sales revenue compared to the original revenue?

\\((2)\\) To expand the influence of the product and increase annual sales, the company plans to conduct a comprehensive technical reform and marketing strategy adjustment for the product next year, raising the price to $\\(x\\)$ yuan. The company plans to invest $\\(\\dfrac{1}{6}({{x}^{2}}-600)\\)$ ten thousand yuan as the technical reform fee, $\\(50\\)$ ten thousand yuan as the fixed promotion fee, and $\\(\\dfrac{1}{5}x\\)$ ten thousand yuan as the variable promotion fee. What is the minimum annual sales volume $\\(a\\)$ required for the product to ensure that next year's sales revenue is not lower than the sum of the original revenue and the total investment? Additionally, find the price per unit of the product under this condition.

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