
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
In response to the national call for "comprehensively implementing the rural revitalization strategy," a certain student decided to start a business using the knowledge gained from university after graduation. After conducting market research, it was found that producing a certain type of small electronic product requires a fixed cost of $30,000. For each production of $x$ thousand units, an additional variable cost $C(x)$ in thousand dollars is required. When the annual output is less than 100,000 units, $C(x)=\frac{9}{x}+7x-14$ (thousand dollars); when the annual output is not less than 100,000 units, $C(x)=6x+lnx+\frac{{4e}}{x}-13$ (thousand dollars). It is known that the selling price of each product is $6. If all the products are sold out in that year, find:

$(1)$ Write the analytical expression of the annual profit $P(x)$ (in thousand dollars) as a function of the annual output $x$ (in thousand units). (Note: Annual profit = Annual sales revenue - Fixed cost - Variable cost)

$(2)$ At approximately how many thousand units of annual output will the product achieve the maximum annual profit? What is the maximum annual profit? (Keep $\ln 2=0.7$)

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