
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
The 31st World University Summer Games will be held in Chengdu, Sichuan from July 28th to August 8th, 2023. A company decided to evaluate a certain product under its umbrella for bidding for related endorsement activities. The original selling price of the product was $25 per unit, with an annual sales volume of 80,000 units.
$(1)$ According to market research, if the price is increased by $1, 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?
$(2)$ To seize this opportunity, expand the influence of the product, and increase the annual sales volume, the company decided to immediately carry out a comprehensive technological innovation and marketing strategy reform on the product, and increase the price to $x per unit. The company plans to invest $\frac{1}{6}(x^{2}-600)$ million as a technological innovation cost, $50$ million as fixed advertising costs, and $\frac{x}{5}$ million as variable advertising costs. What is the minimum sales volume $a$ that the product should reach after the reform in order 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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