
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]


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With the popularization of the concept of "low-carbon living, green travel", new energy vehicles are gradually becoming the favorite means of transportation for the people of Fuqing. It is predicted that the total demand $R(x)$ for BYD cars in the first $x$ months starting from January 2023 at a certain new energy vehicle $4S$ store in Fuqing will approximately satisfy the relationship $R(x) = \frac{1}{2}x(x+1)(39-2x)$ (where $x \in \mathbb{N}^{*}$ and $x \leqslant 6$). The relationship between the purchase price $W(x)$ of this car in the $x$-th month and $x$ is approximately $W(x) = 150000 + 2000x$.  
$(1)$ From the total demand $R(x)$ in the first $x$ months, find the function relationship between the demand $g(x)$ in the $x$-th month and $x$;  
$(2)$ The selling price of each car is $185000$ yuan. If no other expenses are considered, in which month of 2023 will the monthly profit $f(x)$ of this car $4S$ store be maximized, and what is the maximum monthly profit in yuan?

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