
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 city is promoting the construction of beautiful rural areas and the development of agricultural economy. They encourage a food company to produce a beverage that costs $10$ yuan per bottle and is sold for $15$ yuan. The monthly sales volume is $80,000$ bottles.<br/>$(1)$ According to market research, if the price per bottle is increased by $1$ yuan, the monthly sales volume will decrease by $8,000$ bottles. To ensure that the total profit next month is not lower than the total profit of the current month, what is the maximum price per bottle for the beverage?<br/>$(2)$ In order to increase the total profit per month, the company decides to adjust its marketing strategy next month. They plan to set the price per bottle at $x$ yuan $(x\geqslant 16)$ and invest $\frac{{33}}{4}(x-16)$ million yuan as the cost of adjusting the marketing strategy. According to market research, for every $1$ yuan increase in the price per bottle, the monthly sales volume will decrease by $\frac{{0.8}}{{{{(x-15)}^2}}}$ million bottles. What price per bottle $x$ will maximize the total profit next month? Also, calculate the maximum total profit next month. (Hint: Total profit per month $=$ Total sales revenue per month $-$ Total cost per month)

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