
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
Recently, with the continuous fermentation of the Li Jiaqi live streaming incident, domestic brands have staged fancy live broadcasts. A certain brand also wants to take advantage of this trend by adopting methods such as "large quantity, favorable price" and "advertising promotion" to increase the sales of a certain product under its brand. Market research has found that this product is in short supply, and all produced units can be sold. The monthly sales volume $m$ (in ten thousand units) of this product is related to the advertising promotion cost $x$ (in ten thousand yuan) as follows: $m=12-\frac{18}{2x+1}$. The relationship between the unit price $n$ and the sales volume is defined as $n=9+\frac{9}{m}$ thousand yuan. It is known that the cost of producing ten thousand units of this product is 8 thousand yuan. Let the profit of this product be $y$ thousand yuan. <br/> $(1)$ Find the functional relationship between $y$ and $x$; (Profit $=$ Sales revenue $-$ Cost $-$ Advertising promotion cost) <br/> $(2)$ When the advertising promotion cost is set to how many thousand yuan, the profit of this product is maximized? What is the maximum profit in thousand yuan?

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