
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 company decided to evaluate a certain product under its umbrella. The original price of the product was $25$ yuan per unit, with annual sales of $80,000$ units.  
$(1)$ According to market research, if the price is increased by $1$ yuan, the sales volume will decrease by $2,000$ units. To ensure that the total revenue from sales is not less than the original revenue, what is the maximum price per unit that the product can be priced at?  
$(2)$ In order to expand the influence of the product and increase the annual sales volume, the company decided to immediately carry out a comprehensive technological innovation and sales strategy adjustment for the product, and increase the price to $x$ yuan. The company plans to invest $\frac{1}{6}({{x^2}-600})$ million yuan as the cost of technological innovation, invest $50$ million yuan as fixed advertising costs, and invest $\frac{x}{5}$ million yuan as variable advertising costs. What is the minimum sales volume $a$ that the product must reach after the reform in order to ensure that the sales revenue after the reform is not less than the sum of the original revenue and the total investment? Also, determine the price per unit of the product at this time.

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