
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 $25$ dollars per unit, with annual sales of $80,000$ units.  
$(1)$ According to market research, if the price is increased by $1$ dollar, 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 undergo a comprehensive technological innovation and sales strategy adjustment for the product, and increase the price to $x$ dollars. The company plans to invest $\frac{1}{6}({{x^2}-600})$ million dollars as the cost of technological innovation, $50$ million dollars as fixed advertising costs, and $\frac{x}{5}$ million dollars as variable advertising costs. What is the minimum sales volume $a$ for the reformed product, in million units, to ensure that the sales revenue after the reform is not less than the original revenue plus the total investment? Also, determine the price per unit of the product at this point.

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